Parameter Identification for Dispersive Dielectrics Using

Pulsed Microwave Interrogating Signals and Acoustic Wave

Induced Reflections in Two and Three Dimensions

H. T. Banks† and V. A. Bokil‡

Center for Research in Scientific ComputationNorth Carolina State University

Raleigh, N.C. 27695-8205

July 2004

†email: htbanks@ncsu.edu‡email: vabokil@ncsu.edu

Abstract

In this report we consider an electromagnetic interrogation technique for identifying thedielectric parameters of a Debye medium in two and three dimensions. These parametersinclude the dielectric permittivity, the conductivity and the relaxation time of the Debyemedium. In this technique a travelling acoustic pressure wave that is generated in the Debyemedium is used as a virtual reflector for an interrogating microwave electromagnetic pulsethat is generated in free space and impinges on a planar interface that separates air and theDebye medium. The reflections of the microwave pulse from the air-Debye interface and fromthe acoustic pressure wave are recorded at an antenna that is located in air. These reflectionscomprise the data that is used in an inverse problem to estimate the dielectric parametersof the Debye medium. We assume that the dielectric parameters of the Debye mediumare locally pressure dependent. A model for acoustic pressure dependence of the materialconstitutive parameters in Maxwell’s equations is presented. As a first approximation, weassume that the Debye dielectric parameters are affine functions of pressure. We present atime domain formulation that is solved using finite differences in time and in space using thefinite difference time domain method (FDTD). Perfectly matched layer (PML) absorbingboundary conditions are used to absorb outgoing waves at the finite boundaries of thecomputational domain and prevent excessive spurious reflections from reentering the domainand contaminating the data that is collected at the antenna placed in air.

Using the method of least squares for the parameter identification problem we solve an

inverse problem by using two different algorithms; the gradient based Levenberg-Marquardt

method, and the gradient free, simplex based Nelder-Mead method. We solve inverse prob-

lems to construct estimates for two or more dielectric parameters. Finally we use statistical

error analysis to construct confidence intervals for all the presented estimates. These con-

fidence intervals are a probabilistic statement about the procedure that we have used to

construct estimates of the various parameters. Thus we naturally combine the deterministic

nature of our problem with uncertainty aspects of estimates.

2

1 Introduction

The study of transient electromagetic waves in lossy dispersive dielectrics is of greatinterest due to the numerous applications of this subject to many different problems[AMP94, APM89]. These include microwave imaging for medical applications inwhich one seeks to investigate the internal structure of an object (the human body)by means of electromagnetic fields at microwave frequencies (300 MHz - 30 GHz). Thisis done for example, to detect cancer or other anomalies by studying changes in theelectrical properties of tissues [FMS03]. These include the dielectric properties suchas permittivity, the conductivity and the relaxation time for the Debye medium. Onegenerates the electrical property distributions in the body (Microwave maps) with thehope that such properties of different bodily tissues are related to their physiologicalstate. With such noninvasive interrogating techniques one can study properties anddefects in biological tissues with very little discomfort to the subjects. Other potentialapplications for such interrogation techniques are nondestructive damage detectionin aircraft and spacecraft where very high frequency electromagnetic pulses can beused to detect the location and width of any cracks that may be present [BGW03].Additional applications are found in mine, ordinance and camouflage surveillance,and subsurface and atmospheric environmental modeling [BBL00].

Another important technique employed in noninvasive interrogation of objects isthe use of ultrasonic waves in industrial and medical applications. The interaction ofelectromagnetic and sound waves in matter has been widely studied in acoustooptics[And67, Kor97]. Brillouin first showed that electromagnetic and sound waves caninteract in a medium and influence each others propagation dynamics [Bri60]. Theresponse of the atomic electrons in a material medium to the applied electric fieldresults in a polarization of the material with a resulting index of refraction that is afunction of the density in the material. Consequently, the material density fluctua-tions produced by a sound wave induce perturbations in the index of refraction. Thusan electromagnetic wave transmitted through the medium will be modulated by thesound wave, and scattering and reflection will occur. Conversely, a spatial variationof the electromagnetic field intensity and the corresponding electromagnetic stresslead to a volume force distribution in the medium. This is called electrostriction, andcan lead to sound generation. This mutual interaction between light and sound mayalso lead to instabilities and wave amplification [MI68].

The work in this paper is based on ideas underlying the techniques formulated in[BBL00] and [ABR02]. In [BBL00], the authors present an electromagnetic interro-gation technique for general dispersive media backed by either a perfect conductingwall to reflect the electromagnetic waves, or by using standing acoustic waves asvirtual reflectors. In [ABR02], this work was extended to using travelling waves asacoustic reflectors. In both cases the techniques were demonstrated in one dimensionby assuming plane electromagnetic waves impinging at normal incidence on slabs ofdispersive media. In [BB03] a multidimensional version of the electromagnetic in-

3

terrogation problem is presented for dispersive media backed by perfect conductorwalls and the FDTD method is utilized [Taf95, Taf98]. However, as we have notedabove, the object of interrogation may not be backed by perfect conductor walls inmany important applications, and hence one needs to have another type of accessibleinterface which will reflect the propagating electromagnetic pulses. It is thereforeof great importance to ascertain (computationally and experimentally) whether theinterrogation ideas of [ABR02] using acoustic pressure waves as virtual reflectors canbe effective in multidimensional settings. The goal of this paper is the investigationof computational and statistical aspects of this methodology. At present we are alsotrying to validate these proposed techniques by testing the ideas in this paper in anexperimental setup. Our group has built an antenna that we are using to study theacoustooptical interaction in agar-agar in which the sound waves are produced bymeans of a transducer [ABK04]. In such a setup, as well as in many applications onecannot assume normally incident plane waves impinging on the agar, and hence oneneeds to consider the more general case of oblique incidence. This motivates the studyof an electromagnetic interrogation technique using the acoustooptical interaction tostudy dielectic properties of dispersive media in multidimensions.

We begin with Maxwell’s equations for a first order dispersive (Debye) mediawith Ohmic conductivity and incorporate a model that describes the electromag-netic/acoustic interaction. Our model features modulation of the material polariza-tion by the pressure wave and thus the behaviour of the electromagnetic pulse. Thisapproach is based on ideas found in [MI68]. We incorporate a pressure dependentDebye model for orientational polarization into Maxwell’s equations. We model thepropagation of a non-harmonic pulse from a finite antenna source in free space im-pinging obliquely on a planar interface into the Debye medium. We use the finitedifference time domain method (FDTD) [Taf95] to discretize Maxwell’s equationsand to compute the components of the electric field in the case where the signal andthe dielectric parameters are independent of the y variable. Figure 1 depicts theantenna and the Debye dielectric slab geometry that we will model in our problemwith the infinite dielectric slab perpendicular to the z-axis and uniform in the y-direction and finite in the x-direction lying in the region −∞ ≤ y ≤ ∞, x1 ≤ x ≤ x2.An alternating current along the x-direction of the antenna located at z = zc thenproduces an electric field that is uniform in y with nontrivial components Ex andEz depending on (t, x, z). When propagated in the xz-plane, this results in obliqueincident waves on the dielectric surface in the xy-plane at z = z1. (In [BF95] theauthors use Fourier-series in the frequency domain to compute the propagation ofa time harmonic pulse train of plane waves that enter a dielectric across a planarboundary at oblique incidence. They showed that precursors are excited by shortrise time pulses at oblique incidence. However, the use of Fourier series restrictsthis approach to harmonic pulses.) The oblique waves impinge on the planar inter-face that separates air and the Debye medium at z = z1 where they undergo partialtransmission and partial reflection. A reflected wave travels back to the antenna at

4

z = zc and is recorded. The transmitted part of the wave travels through the Debyemedium and intercepts an acoustic pressure wave that is generated in the dielectricmedium. Here the microwave pulse undergoes partial transmission and reflection.The reflected wave travels back through the Debye medium, undergoes a secondaryreflection/transmission at the air-Debye interface and is then recorded at the antenna.Thus the antenna records three pieces of information; the input source, the reflectionof the electromagnetic pulse from the air-Debye interface and the reflection from thepressure wave region. This data is recorded at the center of the antenna and will beused in an inverse problem to estimate the dielectric properties of the Debye medium.We note that the acoustic wave speed is many orders of magnitude smaller than thespeed of light. Thus the acoustic pressure wave is almost stationary relative to themicrowave pulse. In this report, we have assumed that the acoustic pressure waveis specified a priori and hence we assume that the pressure wave is not modified bythe electromagnetic wave. This leads to a much simplified electromagnetic/acousticinteraction model. The dynamic interaction between the two and the appropriatenessof this simplifying assumption are currently being investigated and will be the topicof a future report.

An important computational issue in modeling two or three dimensional wavepropagation is the construction of a finite computational domain and appropriateboundary conditions for terminating this domain in order to numerically simulatean essentially unbounded domain problem. We surround the domain of interest byperfectly matched absorbing layers (PML’s), thus producing a finite computationaldomain. The original PML formulation is due to J. P. Berenger [Ber94] and was basedon a split form of Maxwell’s equations in which all the fields themselves were splitinto orthogonal subcomponents. This formulation was appropriately called the splitfield form. Many different variations of the split field PML are now available in theliterature. Most of these different formulations, though equivalent to the split fieldPML, do not involve the splitting of the fields or splitting of Maxwell’s equations.The PML formulation that we use in this report is based on the anisotropic (uniaxial)formulation that was first proposed by Sacks et. al. [SKLL95].

We first present numerical results for the forward problem for two different testcases. The relaxation times of the Debye media in these two test cases differ bymany orders of magnitude. We perform a sensitivity analysis in each test case todetermine which parameters are most likely to be accurately estimated in an inverseproblem. We also determine which of the pressure dependent parameters are the mostsignificant in each of the test cases. As we shall see, the significant difference in theorders of magnitude of the relaxation times for the two test cases produces inverseproblems with very different estimation properties.

We next discuss a parameter estimation problem in which we estimate modelparameters for the two different Debye media from simulated data. We formulateour inverse problem using the method of least squares. We compare two differentalgorithms in solving the parameter estimation problems; the Levenberg-Marquardt

5

method, which is a gradient based algorithm, and the Nelder-Mead method, which isbased on simplices and is gradient free. For the first test case, these two algorithmsproduce similar results. However, in the second test case, we see the presence of manylocal minima and the Nelder-Mead algorithm converges to a local minima instead ofthe global minimum.

We also consider uncertainty in estimates due to data uncertainty. We presenta model for generating noisy data that we use in our inverse problems to simulateexperimental settings in which noise enters into the problem in a natural fashion. Wethen discuss a statistical error methodology to analyze the results of the inverse prob-lems based on noisy data. This statistical error analysis is used to obtain confidenceintervals for all the parameters estimated in the inverse problem. These intervalsindicate the level of confidence that can be associated with estimates obtained withour methodology.

2 Model formulation: The forward problem

We consider Maxwell’s equations which govern the electric field E and the magneticfield H in a domain Ω with charge density ρ. Thus we consider the system

(i)∂D

∂t+ J −∇× H = 0, in (0, T ) × Ω,

(ii)∂B

∂t+ ∇× E = 0, in (0, T ) × Ω,

(iii) ∇ · D = ρ, in (0, T ) × Ω,

(iv) ∇ · B = 0, in (0, T ) × Ω,

(v) E × n = 0, on (0, T ) × ∂Ω,

(vi) E(0,x) = 0, H(0,x) = 0, in Ω.

(1)

The fields D,B are the electric and magnetic flux densities respectively. All thefields in (1) are functions of position r = (x, y, z) and time t. We have J = Jc + Js,where Jc is a conduction current density and Js is the source current density. Weassume only free space (actually the antenna in our example) can have a sourcecurrent, and Jc is only found in the dielectric material. The condition (1, v) is aperfectly conducting boundary condition on the computational domain which we willreplace with absorbing conditions in the sequel. With perfect conductor conditions,the computational boundary acts like a hard wall to impinging electromagnetic wavescausing spurious reflections that would contaminate the data that we wish to use inour inverse problems.

Constitutive relations which relate the electric and magnetic fluxes D,B and theelectric current Jc to the electric and magnetic fields E,H are added to these equationsto make the system fully determined and to describe the response of a material to the

6

electromagnetic fields. In free space, these constitutive relations are D = ε0E, andB = µ0H, where ε0 and µ0 are the permittivity and the permeability of free space,respectively, and are constant [Jac99]. In general there are different possible formsfor these constitutive relationships. In a frequency domain formulation of Maxwell’sequations, these are usually converted to linear relationships between the dependentand independent quantities with frequency dependent coefficient parameters. We willconsider the case of a dielectric in which magnetic effects are negligible, and we assumethat Ohm’s law governs the electric conductivity. Thus, within the dielectric mediumwe have constitutive relations that relate the flux densities D,B to the electric andmagnetic fields, respectively. We have

(i) D = ε0E + PID,

(ii) B = µ0H,

(iii) Jc = σEID,

(2)

In (2), ID denotes the indicator function on the Debye medium. Thus, Jc = 0 inair. The quantity P is called the electric polarization. It is equal to zero in air andis nonzero in the dielectric. Electric polarization may be defined as the electric fieldinduced disturbance of the charge distribution in a region. This polarization mayhave an instantaneous component as well as ones that do not occur instantaneously;the latter usually have associated time constants called the relaxation times which aredenoted by τ . We let the instantaneous component of the polarization to be relatedto the electric field by means of a dielectric constant ε0χ and denote the remainderof the electric polarization as PR. We have

P = PI + PR = ε0χE + PR, (3)

and hence the constitutive law (2, i) becomes

D = ε0εrE + PR, (4)

where εr = (1 + χ) is the relative permittivity of the dielectric medium.To describe the behaviour of the media’s macroscopic electric polarization PR,

we employ a general integral equation model in which the polarization explicitly de-pends on the past history of the electric field. This model is sufficiently general toinclude microscopic polarization mechanisms such as dipole or orientational polariza-tion(Debye), as well as ionic and electronic polarization(Lorentz), and other frequencydependent polarization mechanisms [And67] as well as multiples of such mechanismsrepresented by a distribution of the associated time constants (e.g., see [BG04]). Theresulting constitutive law can be given in terms of a polarization or displacementsusceptibility kernel g as

PR(t, x, z) =

∫ t

0

g(t − s, x, z)E(s, x, z)ds, (5)

7

1z 1z

x

y

E(t,x,z)

Figure 1: Antenna and dielectric slab configuration.

which, in the case (Debye) of interest to us here, can be written as

g(t) = e−t/τ ε0(εs − ε∞)

τ. (6)

Such a Debye model can be represented in differential form as

τPR + PR = ε0(εs − ε∞)E,

D = ε0ε∞E + PR,(7)

inside the dielectric, whereas PR = 0, ε∞ = 1 in air, and ε∞ = εr in the dielectric.We will henceforth denote PR by P. In equations (5) and (7), the parameters εs

and ε∞ denote the static relative permittivity, and the value of permittivity for anextremely high(≈ ∞) frequency field, respectively. The quantity ε∞ is called the infi-nite frequency permittivity. Biological cells and tissues display very high values of thedielectric constants at low frequencies, and these values decrease in almost distinctsteps as the excitation frequency is increased. This frequency dependence of permit-tivity is called dielectric dispersion and permits identification and investigation of anumber of completely different underlying mechanisms. This property of dielectricmaterials makes the problem of parameter identification an important as well as veryuseful topic for investigation.

We will modify system (1) and the constitutive laws (2) by performing a changeof variables that renders the system in a form that is convenient for analysis andcomputation. From (1, i) we have

∂

∂t

(D +

∫ t

0

Jc(s,x) ds

)−∇×H = −Js, in (0, T ) × Ω. (8)

8

Next, we define the new variable

D(t,x) = D(t,x) +

∫ t

0

Jc(s,x) ds. (9)

Using definition (9) in (8) and (1) we obtain the modified system

(i)∂D

∂t−∇×H = −Js in (0, T ) × Ω,

(ii)∂B

∂t+ ∇×E = 0 in (0, T ) × Ω,

(iii) ∇ · D = 0 in (0, T ) × Ω,

(iv) ∇ · B = 0 in (0, T ) × Ω,

(v) E × n = 0 on (0, T ) × ∂Ω,

(vi) E(0,x) = 0, H(0,x) = 0 in Ω.

(10)

We will henceforth drop the˜ symbol on D. We note that equation (10, iii) followsfrom the continuity equation ∂ρ

∂t+ ∇ · J = 0, the assumption that ρ(0) = 0, and the

assumption that ∇ · Js = 0 (in the sense of distributions). The modified constitutivelaws (2) after substitution of Ohm’s law and (9) are

(i) D(t,x) = ε0ε∞(x)E(t,x) + PR +

∫ t

0

σ(x)E(s,x) ds,

(ii) B = µ0H,(11)

with PR, the solution ofτPR + PR = ε0(εs − ε∞)E. (12)

3 An anisotropic perfectly matched layer absorb-

ing medium

In one dimension exact absorbing boundary conditions can be constructed for ter-minating a computational domain. In two and three dimensions, however, there areno exact absorbing boundary conditions available. In such a case we can either useapproximate absorbing boundary conditions of some finite order (usually one or two),or we can surround our computational domain by absorbing layers. In this report weuse the second approach and construct perfectly matched layers [Ber94] surroundingthe domain of interest. These layers have been demonstrated to have many orders ofmagnitude better absorbing capabilities than the approximate absorbing boundaryconditions [Taf98]. In this section we will first construct PML’s that can be matched

9

to free space and then we will explain how this construction can be generalized to thecase of a Debye medium.

The source current Js is zero inside the PML regions. So we will neglect Js inthis section. We model the PMLs as lossy regions by adding artificial loss terms toMaxwell’s equations. This approach is based on using anisotropic material propertiesto describe the absorbing layer [SKLL95, Ged96, Rap95]. We start with Maxwell’sequations in the most general form

∂D

∂t= ∇×H − Je,

∂B

∂t= −∇×E − Jm,

∇ · D = 0,

∇ · B = 0,

(13)

where Je and Jm are electric and magnetic conductivities, respectively. Inside thecomputational domain we will have Je = Jc and Jm = 0. We will derive a PMLmodel in the frequency domain and then transform the corresponding equations tothe time domain by taking the inverse Fourier transforms of the frequency domainequations. To this end, we consider the time-harmonic form of Maxwell’s equations(13) given by

jωD = ∇×H − Je,

jωB = −∇×E − Jm,

∇ · B = 0,

∇ · D = 0,

(14)

where for every field vector V, V denotes its Fourier transform. We then have theconstitutive laws

B = [µ]H,

D = [ε]E,

Jm = [σM ]H,

Je = [σE]E.

(15)

In (15) the square brackets indicate a tensor quantity. Thus we allow for anisotropy inour media by permitting the material parameters to be tensors. Note that when thedensity of electric and magnetic charge carriers in the medium is uniform throughoutspace, then ∇ · Je = 0 and ∇ · Jm = 0. We define the new tensors

[µ] = [µ] +[σM ]

jω,

[ε] = [ε] +[σE]

jω.

(16)

10

Using the definitions (16) we define two new constitutive laws that are equivalent to(15), given by

Bnew = [µ]H,

Dnew = [ε]E.(17)

Using (17) in (14), we can write Maxwell’s equations in time-harmonic form as

jωDnew = ∇×H,

jωBnew = −∇×E,

∇ · Dnew = 0,

∇ · Bnew = 0.

(18)

The split-field PML introduced by Berenger [Ber94] is a hypothetical mediumbased on a mathematical model. In [MP95] Mittra and Pekel showed that Berenger’sPML was equivalent to Maxwell’s equations with a diagonally anisotropic tensorappearing in the constitutive relations for D and B. For a single interface, theanisotropic medium is uniaxial and is composed of both the electric and magneticpermittivity tensors. This uniaxial PML (UPML) formulation performs as well asthe original split-field PML while avoiding the nonphysical field splitting. As will beshown below, by properly defining a general constitutive tensor [S], we can use theUPML in the interior working volume as well as the absorbing layer. This tensorprovides a lossless isotropic medium in the primary computation zone, and individualUPML absorbers adjacent to the outer lattice boundary planes for mitigation ofspurious wave reflections. The fields excited within the UPML are also plane wave innature and satisfy Maxwell’s curl equations. The derivation of the PML propertiesof the tensor constitutive laws is also done directly by Sacks et al. in [SKLL95] andby Gedney in [Ged96]. We follow the derivation by Sacks et al. here. We beginby considering planar electromagnetic waves in free space incident upon a PML halfspace. Starting with the impedance matching assumption, i.e., the impedance of thelayer must match that of free space: ε−1

0 µ0 = [ε]−1[µ] we have

[ε]

ε0

=[µ]

µ0

= [S] = diaga1, a1, a3. (19)

Hence, the constitutive parameters inside the PML layer are [ε] = ε0[S] and [µ] =µ0[S], where [S] is a diagonal tensor. We next consider plane wave solutions of theform

V(x, t) = V(x) exp(j(ωt − k · x)), (20)

for all field vectors V, to the time-harmonic Maxwell’s equations with the diagonallyanisotropic tensor, where k = (kx, ky, kz) is the wave vector of the planar electromag-netic wave and x = (x, y, z). The dispersion relation for waves in the PML are found

11

to bek2

x

a2a3

+k2

y

a1a3

+k2

z

a1a2

= k20 ≡ ω2µ0ε0 ≡

ω2

c20

. (21)

where, c0 is the speed of light in free space.To further explain our UPML formulation, we may, without loss of generality,

consider a PML layer which fills the positive x half-space and plane waves with wavevectors in the xz- plane (ky = 0). Let θi be the angle of incidence of the plane wavemeasured from the normal to the surface x = 0. The standard phase and magnitudematching arguments at the interface yield the following generalization of Snell’s law

√a1a3 sin θt = sin θi, (22)

where θt is the angle of the transmitted plane wave. Matching the magnitudes of theelectric and magnetic fields at the interface, x = 0, we obtain the following values ofthe reflection coefficients for the TE and the TM modes

RTE =

cos θi −√

a3

a2

cos θt

cos θi +

√a3

a2

cos θt

RTM =

√a3

a2

cos θt − cos θi

cos θi +

√a3

a2

cos θt

(23)

From (23) we can see, that by choosing a3 = a2 = a and√

a1a3 = 1, the interfaceis completely reflectionless for any frequency, angle of incidence, and polarization.Using (17) and (19), we thus find the constitutive laws for the perfectly matchedlayer are

Dnew = ε0[S]E,

Bnew = µ0[S]H,(24)

where the tensor [S] is

[S] =

a−1 0 00 a 00 0 a

. (25)

The perfectly matched layer is therefore characterized by the single complex numbera. Taking it to be the constant a = γ − jβ, and substituting into the dispersionrelation (21), we obtain the following expression for the electric field inside of thePML

E(x, y, z) = E0 exp(−k0β cos θtx) exp(−jk0(γ cos θtx + sin θtz)) exp(jωt). (26)

Hence we can see that γ determines the wavelength of the wave in the PML and forβ > 0, the wave is attenuated according to the distance of travel in the x direction.

12

Remark 1 To match the anisotropic PML along a planar boundary to a dispersiveisotropic half-space, we will need to modify the above construction for the phasor-domain constitutive relation

D = ε0εr(ω)E, (27)

where εr(ω) is the frequency dependent relative permittivity of the Debye medium,which will be defined in a standard manner in (40) below.

3.1 Implementation of the uniaxial PML

To apply the perfectly matched layer to electromagnetic computations, we replace thehalf infinite layer by a layer of finite depth backed with a more conventional boundarycondition, such as a perfect electric conductor (PEC). This truncation of the layer willlead to reflections generated at the PEC surface, which can propagate back throughthe layer to re-enter the computational region. In this case, the reflection coefficientR, is now a function of the angle of incidence θ, the depth of the PML δ, as well asthe parameter a in (25). Thus, this parameter a for the PML is chosen in order forthe attenuation of waves in the PML to be sufficient so that the waves striking thePEC surface are negligible in magnitude. Perfectly matched layers are then placednear each edge (or face in the 3D case) of the computational domain where a non-reflecting condition is desired. This leads to overlapping PML regions in the cornersof the domain. As shown in [SKLL95], the correct form of the tensor which appearsin the constitutive laws for these regions is the product

[S] = [S]x[S]y[S]z, (28)

where component [S]α in the product (28) is responsible for attenuation in the αdirection, for α = x, y, z (see Figure 2). All three of the component tensors in (28)are diagonal and have the forms

[S]x =

s−1x 0 00 sx 00 0 sx

; [S]y =

sy 0 00 s−1

y 00 0 sy

; [S]z =

sz 0 00 sz 00 0 s−1

z

. (29)

In the above sx, sy, sz are analogous to the complex valued parameter a encounteredin the Section 3 analysis of the single PML layer. Thus, sα governs the attenuation ofthe electromagnetic waves in the α direction for α = x, y, z. When designing PML’sfor implementation, it is important to choose the parameters sα so that the resultingfrequency domain equations can be easily converted back into the time domain. Thesimplest of these [Ged96] which we employ here, is

sα = 1 +σα

jωε0

, where σα ≥ 0, α = x, y, z. (30)

13

Working Volume

- [S] = [S]z

6

?

[S] = [S]x

PML PEC WALL

?Corner Region

?

6

?6?

6?

σx = 0

σx > 0

σx > 0

-- -σz = 0 σz > 0σz > 0

@@

@R

Figure 2: PML layers surrounding the domain of interest. In the corner regions of thePML, both σx and σz are positive and the tensor [S] is the product [S]x[S]z. In theremaining regions only one of σz (left and right PML’s) or σx (top and bottom PML’s)are nonzero and positive. The tensor [S], is thus either [S]x or [S]z respectively. ThePML is truncated by a perfect electric conductor (PEC).

The PML interface represents a discontinuity in the conductivities σα. To reducethe numerical reflections caused by these discontinuous conductivities, the σα arechosen to be functions of the variable α (e.g., σx is taken to be a function of x inthe [S]x component of the PML tensor). A choice of these functions so that σα = 0,i.e., sα = 1 at the interface yields the PML a continuous extension of the mediumbeing matched and reduces numerical reflections at the interface. If one increasesthe value of σα with depth in the layer, one obtains greater overall attenuation whileminimizing the numerical reflections. Gedney [Ged96] suggests a conductivity profile

σα(α) =σmax|α − α0|m

δm; α = x, y, z. (31)

where δ is the depth of the layer, α = α0 is the interface between the PML andthe computational domain, and m is the order of the polynomial variation. Gedneyremarks that values of m between 3 and 4 are believed to be optimal. For theconductivity profile (31), the PML parameters can be determined for given values ofm, δ, and the desired reflection coefficient at normal incidence R0, as

σmax ≈ (m + 1) ln(1/R0)

2ηIδ, (32)

ηI being the characteristic wave impedance of the PML. Emperical testing suggests

14

z1 z2z2 + λp

ZRpml

ZRpml

Ω :

z = zc

ZLpml Za ZD

ZLpml Za ZD

XBpml

X0

XTpml

XBpml

X0

XTpml

x1

x2

Antenna

x

zy

6-

@@R

PEC wall

Figure 3: PML layers surrounding the domain of interest.

that, for a broad range of problems, an optimal value of σmax is given by

σopt ≈m + 1

150π∆α√

εr

, (33)

where ∆α is the space increment in the α direction and εr is the relative permittivityof the material being modeled. In the case of free space εr = 1. Since we desire tomatch the PML to both free space as well as the Debye medium, we will use εr to bethe average value

εr =1

2(1 + ε∞) . (34)

3.2 Reduction to two dimensions

From the time-harmonic Maxwell’s curl equations in the PML (18) and (24), Ampere’sand Faraday’s laws can be written in the most general form as

jωµ0[S]H = −∇×E, (Faraday’s Law)

jω[S]D = ∇×H − Js, (Ampere’s Law)(35)

In (35), [S] is the diagonal tensor defined via (28), (29) and (30). In the presence ofthis diagonal tensor, a plane wave is purely transmitted into the uniaxial medium.The tensor [S] is no longer uniaxial by strict definition, but rather is anisotropic.However, the anisotropic PML is still referenced as uniaxial since it is uniaxial inthe nonoverlapping PML regions. Let Ω = X × Z denote the computational domainalong with the absorbing layers. We partition the interval X into disjoint closed

15

intervals as X = XBpml ∪X0 ∪XTpml and the interval Z into disjoint closed intervalsas ZLpml∪Za∪ZD∪ZRpml, as seen in Figure 3. The computational domain of interestis the region X0 × Za ∪ ZD, where X0 × Za denotes air and X0 × ZD denotes theDebye medium. The buffer region D \ (X0 × Za ∪ ZD) contains the absorbinglayers (PML’s). The PML’s are backed by a perfect conductor where the boundarycondition n × E = 0 is used. We construct perfectly matched layers in regionswhere (x, z) ∈ (XBpml ∪ XTpml) × (ZLpml ∪ ZRpml). To obtain a two-dimensionalmodel, we make the assumption that the signal and the dielectric parameters areindependent of the y variable. Also, an alternating current along the x direction, asin (54), then produces an electric field that is uniform in y with nontrivial componentsEx and Ez depending on (t, x, z) which, when propagated in the xz plane result inoblique incident waves on the dielectric surface in the xy plane at z = zc. With theseassumptions, Maxwell’s equations reduces to a two dimensional system for solutionsinvolving the Ex and Ez components for the electric field and the component Hy forthe magnetic field, which is refered to as the TEy mode. In this case, we have σy = 0and sy = 1 in the UPML. From (11, i) and (7) expressed in the frequency domain wehave the constitutive relation

D = ε0

(ε∞ +

εs − ε∞1 + jωτ

+σ

jωε0

)E. (36)

Rescaling the electric field as˜E =

√ε0

µ0

E, (37)

we can write the time-harmonic frequency domain Maxwell’s equations (35) in theuniaxial medium as

(i) jω

(1 +

σz(z)

jωε0

)(1 +

σx(x)

jωε0

)−1ˆDx = −c0(

∂Hy

∂z+ Js · x),

(ii) jω

(1 +

σx(x)

jωε0

)(1 +

σz(z)

jωε0

)−1ˆDz = c0

∂Hy

∂x,

(iii) jω

(1 +

σx(x)

jωε0

)(1 +

σz(z)

jωε0

)Hy = c0

(∂Ez

∂x− ∂Ex

∂z

).

(38)

In the above c0 = 3.0 × 108 m/s is the speed of light in vacuum and ˆD = ( ˆDx,

ˆDz)T

is defined asˆD = ε∗r · ˆ

E, (39)

with

ε∗r = ε∞ +εs − ε∞1 + jωτ

+σ

jωε0

. (40)

To avoid a computationally intensive implementation, we define suitable constitutiverelationships that facilitate the decoupling of the frequency dependent terms [Taf98].

16

To this end, we introduce the fields

D∗x = s−1

xˆDx,

D∗z = s−1

zˆDz,

H∗y = szHy.

(41)

Again, dropping the˜symbol on D and transforming the frequency domain equationsto the time domain, we obtain the system

(i) ∂tH∗y +

σx(x)

ε0

H∗y = c0

(∂Ez

∂x− ∂Ex

∂z

),

(ii) ∂tHy +σz(z)

ε0

Hy = ∂tH∗y ,

(iii) ∂tD∗x +

σz(z)

ε0

D∗x = −c0(

∂Hy

∂z+ Js · x),

(iv) ∂tDx = ∂tD∗x +

σx(x)

ε0

D∗x,

(v) ∂tD∗z +

σx(x)

ε0

D∗z = c0

∂Hy

∂x,

(vi) ∂tDz = ∂tD∗z +

σz(z)

ε0

D∗z ,

(42)

withD(t) = ε∞E(t) + P(t) + C(t), (43)

where D = (Dx, Dz)T , and inside of the dielectric the polarization P = (Px, Pz)

T

satisfiesτP + P = (εs − ε∞)E, (44)

while the conductivity term C = (Cx, Cy)T satisfies

C = (σ/ε0)E . (45)

inside of the dielectric. Outside the dielectric we have P = C = 0. We will assumezero initial conditions for all the fields.

3.3 Pressure dependence of polarization

We consider a pressure dependent Debye model for orientational polarization firstdeveloped in [ABR02]. We assume that the material dependent parameters in thedifferential equation (44) for the Debye medium depend on pressure p, i.e.,

τ(p)P + P = (εs(p) − ε∞(p))E. (46)

17

We suppose as a first approximation that each of the pressure dependent parameterscan be prepresented as a mean value plus a perturbation that is proportional to thepressure

(i) τ(p) = τ0 + κτp,

(ii) εs(p) = εs,0 + κsp,

(iii) ε∞(p) = ε∞,0 + κ∞p.

(47)

This can be considered as a truncation after the first-order terms of power seriesexpansion of these functions of p. Thus (44) is written as

(τ0 + κτp)P + P = ((εs,0 − ε∞,0) + (κs − κ∞)p)E. (48)

This model features modulation of the material polarization by the pressure wave andthus the behaviour of the electromagnetic pulse. We will assume that the acousticpressure wave is specified a priori. The pressure p will be assumed to have the form

p(t, z) = I(z2,z2+λp) sin

(ωp[t +

z − z2

cp

]

), (49)

where z2 ∈ ZD with z1 < z2. The terms ωp, λp and cp denote the acoustic frequency,wavelength and speed respectively. We note that outside the pressure region, thepressure coefficients κτ , κs and κ∞ are all zero and

τ = τ0, εs = εs,0, ε∞ = ε∞,0. (50)

From (43) (i) Dx(t) = (ε∞,0 + κ∞p)Ex(t) + Px(t) + Cx(t),

(ii) Dz(t) = (ε∞,0 + κ∞p)Ez(t) + Pz(t) + Cz(t),(51)

with (i) (τ0 + κτp)Px + Px = ((εs,0 − ε∞,0) + (κs − κ∞)p)Ex,

(ii) (τ0 + κτp)Pz + Pz = ((εs,0 − ε∞,0) + (κs − κ∞)p)Ez,(52)

and (i) Cx = (σ/ε0)Ex,

(ii) Cz = (σ/ε0)Ez.(53)

3.4 Source term

The source term Js will model an infinite (in the y direction) antenna strip, finite inthe x direction (between x1 and x2), lying in the z = zc plane in free space, and weassume the signal is polarized with oscillations in the xz plane only, with uniformity

18

in the y direction (see again Figure 1). We therefore take

Js(t, x, z) = I(x1,x2)(x)δ(z − zc) sin(ωc(t − 3t0)) exp

(−[t − 3t0

t0

]2)

x,

t0 =2

ωd − ωc

, ωc = 2πfc rad/sec.

(54)

where I(x1,x2) is the indicator function on the interval (x1, x2) ∈ X0, δ(z − zc) isthe Dirac measure centered at z = zc, and x is the unit vector in the x direction.The Fourier spectrum of this pulse has even symmetry about the frequency fc. Thepulse is centered at time step 3t0 and has a 1/e characteristic decay of t0 time-steps[Taf95]. We will specify the values of all the different quantities involved in (54)when we define the particular problems that we will solve. The source (54) radiatesnumerical waves having time waveforms corresponding to the source function. Inthe discretized model, described below the radiated numerical wave will propagate tothe Debye medium and undergo partial transmission and partial reflection, and timestepping can be continued until all transients decay.

3.5 Space and time discretization

We will use the FDTD algorithm [Taf95] to discretize Maxwell’s equations. TheFDTD algorithm uses forward differences to approximate the time and spatial deriva-tives. The FDTD technique rigorously enforces the vector field boundary conditionsat interfaces of dissimilar media at the time scale of a small fraction of the impingingpulse width or carrier period. This approach is very general and permits accuratemodeling of a broad variety of materials ranging from living human tissue to radarabsorbers to optical glass. The system of equations (42) can be discretized on the stan-dard Yee lattice. The domain Ω is divided into square cells of length h = ∆x = ∆z,h being the spatial increment. The degrees of freedom on an element are shown inFigure 4. The electric field degrees of freedom are at the midpoints of edges of thesquares and the magnetic degrees of freedom are at the centers of cells. The timeinterval over which time stepping is done is divided into subintervals of equal sizeusing the time increment ∆t. We perform normal leapfrogging in time and the lossterms are averaged in time. This leads to an explicit time stepping scheme. In thiscase, a stability condition (CFL) has to be satisfied [Taf95] in order to obtain a wellposed computational scheme. The time step ∆t and the spactial increment h for theFD-TD scheme in non-dispersive dielectics are related by the condition

ηCN =c0∆t

h<

1√2

(55)

The number ηCN is called the Courant number. We fix the value of ηCN = 1/2.In [Pet94], the author established that several extensions of the FD-TD for Debye

19

Ez

Ez

Ex ExHy

Edge 1

Edge 2

Edge

3

Edge

4

h -

h

?

6

@@

@@

@@

@@

-

-

6 6v

Figure 4: Degrees of freedom of E and H on a rectangle.

media satisfy the stability restriction of the standard FD-TD scheme in nondispersivedielectrics. He has also shown that the extensions do not preserve the non-dissipativecharacter of the standard FD-TD scheme. The extended difference systems are moredispersive than the standard FD-TD, and their accuracy depends strongly on howwell the chosen timestep resolves the shortest timescale in the problem regardless ofwhether it is the incident pulse timescale, the medium relaxation timescale, or themedium resonance timescale.

For any field variable V , we use the notation V rl,m = V (r∆t, lh,mh), where r, l,m ∈

N or r = r0±1/2, l = l0±1/2,m = m0±1/2, with r0, l0,m0 ∈ N. With the degrees offreedom as shown in Figure 4, the Maxwell curl equations are discretized as follows.Given the values of the electric field at time n∆t and the values of the magnetic fieldat time (n−1/2)∆t, we then evaluate the magnetic field at time (n+1/2)∆t and theelectric field at time (n + 1)∆t. We have for H∗

y = H∗

H∗|n+1/2i+1/2,k+1/2 − H∗|n−1/2

i+1/2,k+1/2

∆t+

σx(i + 1/2)

ε0

(H∗|n+1/2

i+1/2,k+1/2 + H∗|n−1/2i+1/2,k+1/2

2

)

= c0

(Ez|ni+1,k+1/2 − Ez|ni,k+1/2

∆x

)−c0

(Ex|ni+1/2,k+1 − Ex|ni+1/2,k

∆z

).

(56)

20

Solving (56) for H∗|n+1/2i+1/2,k+1/2 we have the update equation

H∗|n+1/2i+1/2,k+1/2 =

(2ε0 − σx(i + 1/2)∆t

2ε0 + σx(i + 1/2)∆t

)H∗|n−1/2

i+1/2,k+1/2+

(2ε0c0∆t

2ε0 + σx(i + 1/2)∆t

)×

(Ez|ni+1,k+1/2 − Ez|ni,k+1/2

∆x−

Ex|ni+1/2,k+1 − Ex|ni+1/2,k

∆z

).

(57)

Similarly we discretize the update equations for the other field variables in (42). Theupdate equation for Hy = H is

H|n+1/2i+1/2,k+1/2 =

(2ε0 − σz(k + 1/2)∆t

2ε0 + σz(k + 1/2)∆t

)H|n−1/2

i+1/2,k+1/2

+

(2ε0

2ε0 + σz(k + 1/2)∆t

)×(H∗|n+1/2

i+1/2,k+1/2 − H∗|n−1/2i+1/2,k+1/2

).

(58)

The update equation for D∗x is

D∗x|n+1

i+1/2,k =

(2ε0 − σz(k)∆t

2ε0 + σz(k)∆t

)D∗

x|ni+1/2,k

−(

2ε0c0∆t

2ε0 + σz(k)∆t

)×(

Hy|n+1/2i+1/2,k+1/2 − Hy|n+1/2

i+1/2,k−1/2

∆z+ Jn

s,i+1/2,k

),

(59)

where the electromagnetic input source is chosen to have the form Jns,i+1/2,k = Jn

s,i+1/2,kx.The update equation for Dx is

Dx|n+1i+1/2,k = Dx|ni+1/2,k +

(2ε0 + σx(i + 1/2)∆t

2ε0

)D∗

x|n+1i+1/2,k

+

(2ε0 − σx(i + 1/2)∆t

2ε0

)D∗

x|ni+1/2,k.

(60)

The update equation for D∗z is

D∗z |n+1

i,k+1/2 =

(2ε0 − σx(i)∆t

2ε0 + σx(i)∆t

)D∗

z |ni,k+1/2

+

(2ε0c0∆t

2ε0 + σx(i)∆t

)×(

Hy|n+1/2i+1/2,k+1/2 − Hy|n+1/2

i−1/2,k+1/2

∆x

).

(61)

Finally the update equation for Dz is

Dz|n+1i,k+1/2 = Dz|ni,k+1/2 +

(2ε0 + σz(k + 1/2)∆t

2ε0

)D∗

z |n+1i,k+1/2

+

(2ε0 − σz(k + 1/2)∆t

2ε0

)D∗

z |ni,k+1/2.

(62)

21

For the update of Ex and Ez we will combine equations (51)-(53). From (53) we havethe update for Cx and Cz as

Cx|n+1 = Cx|n +

(σ∆t

2ε0

)(Ex|n+1 + Ex|n) ,

Cz|n+1 = Cz|n +

(σ∆t

2ε0

)(Ez|n+1 + Ez|n) .

(63)

From (52) we have the update for Px as

Px|n+1 =

(1 +

∆t

2(τ0 + κτpn+1)

)−1(1 − ∆t

2(τ0 + κτpn)

)Px|n

+

(1 +

∆t

2(τ0 + κτpn+1)

)−1(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn)

2(τ0 + κτpn)

)Ex|n

+

(1 +

∆t

2(τ0 + κτpn+1)

)−1(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn+1)

2(τ0 + κτpn+1)

)Ex|n+1.

(64)

Similarly, from (52) we have the update for Pz as

Pz|n+1 =

(1 +

∆t

2(τ0 + κτpn+1)

)−1(1 − ∆t

2(τ0 + κτpn)

)Pz|n

+

(1 +

∆t

2(τ0 + κτpn+1)

)−1(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn)

2(τ0 + κτpn)

)Ez|n

+

(1 +

∆t

2(τ0 + κτpn+1)

)−1(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn+1)

2(τ0 + κτpn+1)

)Ez|n+1.

(65)

From (51) we have

Ex|n+1 =1

(ε∞,0 + κ∞pn+1)

(Dx|n+1 − Px|n+1 − Cx|n+1

). (66)

Substituting the update equations for P n+1x and Cn+1

x in (66) we obtain an expressionfor En+1

x in terms of Enx , Dn+1

x , Cnx and P n

x given by

F(pn+1) =

(1 +

∆t

2(τ0 + κτpn+1)

)−1(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn+1)

2(τ0 + κτpn+1)

)

+ (ε∞,0 + κ∞pn+1) +σ∆t

2ε0

Ex|n+1 = F(pn+1)−1

[Dx|n+1 − Cn

x − σ∆t

2ε0

Ex|n −(

1 +∆t

2(τ0 + κτpn+1)

)−1

×(

1 − ∆t

2(τ0 + κτpn)

)Px|n +

(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn)

2(τ0 + κτpn)

)Ex|n

],

(67)

22

where Dx|n+1 is calculated from (60). Thus, all the terms on the right side are knownquantities, and we can use equation (67) to update Ex. Similarly the update equationfor Ez is

Ez|n+1 = F(pn+1)−1

[Dz|n+1 − Cn

z − σ∆t

2ε0

Ez|n −(

1 +∆t

2(τ0 + κτpn+1)

)−1

×(

1 − ∆t

2(τ0 + κτpn)

)Pz|n +

(∆t((εs,0 − ε∞,0) + (κs − κ∞)pn)

2(τ0 + κτpn)

)Ez|n

],

(68)

with F(pn+1) as defined in (67). As before, we calculate Dz|n+1 from (62). Hence allterms on the right hand side are known.

4 The forward problem for a Debye medium: First

test case

As discussed in Section 3.3 the pressure dependent parameters ε∗s, ε∗∞, and τ ∗ arerepresented as a mean value plus a perturbation that is proportional to the pressure inequations (47) (as discussed in [ABR02]). In our first test case, we present numericalsimulations for a Debye medium that is similar to water. For signal bandwidths inthe microwave regime the dispersive properties of pure water are usually modeled bya Debye equation having a single molecular relaxation term. The mean values ε∗s,0,ε∗∞,0, and τ ∗

0 for this Debye model [APM89] are given by

ε∗s,0 = 80.1 (relative static permittivity),

ε∗∞,0 = 5.5 (relative high frequency permittivity),

τ ∗0 = 8.1 × 10−12 seconds,

σ∗ = 1 × 10−5 mhos/meter,

(69)

where the ∗ superscript will denote the true values of all the corresponding dielectricparameters. Since we do not yet have experimental data to determine the actualvalues, we instead choose trial values for the coefficients of pressure, κ∗

s, κ∗∞ and κ∗

τ .As a first approximation we take these trial values to be a fraction of the mean valuesof the dielectric parameters, ε∗s,0, ε∗∞,0, and τ ∗

0 , respectively. Thus, in the pressureregion we choose

κ∗s = 0.6ε∗s,0 = 48.06,

κ∗∞ = 0.8ε∗∞,0 = 4.4,

κ∗τ = 0.05τ ∗

0 = 4.05 × 10−13.

(70)

We hope to determine the values of these pressure coefficients from appropriatelydesigned experiments in the near future. We have assumed that the effect of the

23

electric field on the acoustic pressure is negligible. We are interested in how andto what extent the acoustic pressure can change the reflected electric wave fromthe interface at z = z2. Moreover, the relaxation parameter is a characteristic ofthe material in [z1, z2], which affects the transmission of electric waves through thisregion. Consequently, we will examine in more detail how τ0 can effect the electricfield. The electromagnetic input source has the form

Js(t, x, z) = I(x1,x2)δ(z) sin(ωc(t − 3t0)) exp

(−[t − 3t0

t0

]2)

x

t0 =1

2π × 109; ωc = 6π × 109 rad/sec; fc = 3.0 × 109 Hz.

(71)

The Fourier spectrum of this pulse has even symmetry about 3.0 GHz. The com-putational domain is defined as follows. We take X0 = (0, 0.1), Za = (0, 0.15) andZD = (0.15, 0.2). The number of nodes along the z-axis is taken to be 320 and thenumber of nodes along the x-axis is taken to be 160. The spatial step size in boththe x and z directions is ∆x = ∆z = h = 0.1/160. From the CFL condition (55)with the Courant number ηCN = 1/2 we obtain the time increment to be ∆t ≈ 1.0417pico seconds. The central frequency of the input source as described in (71) is 3.0GHz and based on the speed of light in air, c0 = 3× 108 m/s, we calculate the corre-sponding central wavelength to be λc = (2πc0)/ω = 0.1 meters. The antenna is half awavelength long and is placed at (x1, x2)×zc, with zc = 0, x1 = 0.025 and x2 = 0.075.We use PML layers that are half a wavelength thick on all fours sides of the compu-tational domain as shown in Figure 3. The reflections of the electromagnetic pulseat the air-Debye interface and from the acoustic pressure wave are recorded at thecenter of the antenna (xc, zc), with xc = 0.05, at every time step. This data will beused as observations for the parameter identification problem to be presented later.The component of the electric field that is of interest here is the Ex component. Thusour data is the set

E(q∗) = Ex(n∆t, xc, zc;q∗)M

n=1

q∗ = (ε∗s,0, ε∗∞,0, τ

∗0 , σ∗, κ∗

s, κ∗∞, κ∗

τ )T .

(72)

The windowed acoustic pressure wave as defined in (49) has the parameter valuesωp = 6.0π × 105 rad/sec, cp = 1500 m/s, and thus, λp = 0.005. The location of thepressure region is in the interval (z2, z2 + λp) = (0.175, 0.18).

In Figure 5 we plot the electromagnetic source that is used in the simulations forthe Debye model (69). We plot the power spectral density of the source (71) in Figure6. The power spectral density |Y |2 of the vector Js is defined to be

Z = FFT(Js),

|Y |2 = Z · Z,(73)

24

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−9

−200

−150

−100

−50

0

50

100

150

200

Time t (secs)

Sou

rce

(Vol

ts/m

eter

)

Figure 5: Forward Simulation for a Debye medium with parameters given in (69) and(70). The source as defined in (71)

where FFT(Js) is the fast fourier transform of the vector Js. As seen in Figure 6 thepower spectral density is symmetric about 3.0 GHz.

In Figures 7 and 8 we plot the Ex field magnitude at the center of the antennaversus time, which shows the electromagnetic source, the reflection off the air-Debyeinterface and the reflection from the region containing the acoustic pressure wave.As seen in these plots the amplitude of the reflection from the acoustic pressure ismany orders of magnitude smaller than that of the initial electromagnetic source aswell as that of the reflection from the air-Debye interface. Thus, it is an interestingquestion as to whether the reflections from the acoustic pressure region can be usedfor identification of parameters.

In Figure 9 we plot the power spectral density of the data in Figure 7. In Fig-ures 10 -14 we plot the Ex field magnitude in the plane containing the center of theantenna versus depth along the z axis. Figures 10 and 11 shows the electromagneticwave penetrating the Debye medium. In 12 we see the reflection of the electromag-netic wave from the Debye medium moving towards the antenna and the Brillouinprecursor propagating in the Debye medium. In Figure 13 we observe the reflectionfrom the region containing the acoustic pressure and the transmitted part of theelectromagnetic source travelling into the absorbing layer. The reflection from theacoustic pressure crosses the Debye medium in Figure 14.

25

0 3 5 7 9 11 13 15

x 109

0.0

1.0

2.0

3.0

4.0

5.0

Frequency (Hz)

|Y|2

The power spectral density of the Source× 103

Figure 6: Fourier transform of the source. The transform is centered around thecentral frequency of ω = 6π × 109.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−9

−100

−50

0

50

100

150

Time t (secs)

Debye InterfaceReflection

Acoustic Interface Reflection

PSfrag replacements

Ex

fiel

dm

agnitude

(volts/

met

er)

Figure 7: Forward simulation for a Debye medium with parameters given in (69) and(70). Time plot of the Ex component of the electric field. This is the data that isreceived at the center of the antenna. This plot consists of the original signal (source),the reflection off the Air-Debye interface and the reflection due to the pressure wave.

26

1 1.5 2 2.5 3 3.5 4

x 10−9

−30

−20

−10

0

10

20

30

40

Time t (secs)

Debye InterfaceReflection

Acoustic Interface Reflection

PSfrag replacements

Ex

fiel

dm

agnitude

(volts/

met

er)

Figure 8: Forward simulation for a Debye medium with parameters given in (69)and (70). Time plot of the Ex component of the electric field. This is the data thatis received at the center of the antenna. This plot consists of the reflection off theAir-Debye interface and the reflection due to the pressure wave.

0 3 5 7 9 11 13 15

x 109

0.0

1.0

2.0

3.0

4.0

5.0

Frequency (Hz)

|Y|2

The power spectral density of the observed Ex field

× 103

Figure 9: Fourier transform of the data in Figure 7. The transform is centered aroundthe central frequency of ω = 6π × 109.

27

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 7.29 × 10−10 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

Figure 10: Forward simulation for a Debye medium with parameters given in (69)and (70). Plot of the magnitude of the Ex component of the electric field against z(meters). The input source is propagating towards the Debye medium.

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 1.04 × 10−9 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

Figure 11: Forward simulation for a Debye medium with parameters given in (69)and (70). Plot of the magnitude of the Ex component of the electric field against z(meters). The input signal gets partially reflected off the Debye medium and partiallytransmitted into the medium.

28

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 1.25× 10−9 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 1.46 × 10−9 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

Figure 12: Forward simulation for a Debye medium with parameters given in (69)and (70). Plots of the magnitude of the Ex component of the electric field against z(meters). In both the plots, the reflection off the air-Debye interface is seen movingtowards the antenna, while the transmitted part of the source is seen propagating inthe Debye medium towards the region containing the pressure wave.

29

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 2.08 × 10−9 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

Figure 13: Forward simulation for a Debye medium with parameters given in (69)and (70). Plot of the magnitude of the Ex component of the electric field against z(meters). The interaction of the electromagnetic wave with the pressure wave causesthe electromagnetic wave to partially reflect and partially transmit.

4.1 Sensitivity analysis

Here we examine the system dynamics as the parameters vary. We are interested inthe changes produced by the coefficents of pressure in the polarization, namely theparameters κs, κ∞ and κτ . In Figure 15 we plot the Ex component of the electric fieldfor the simulation with parameter values given in (69) and four other simulations forwhich all the parameters, except κs, are fixed at the values given in (69). We changethe value of κs to 0, 0.3εs,0, 0.6εs,0 and 0.9εs,0, repectively, and plot the electric fieldvalues for each of the corresponding forward simulations. As expected, by changingthe value of κs, we notice a change in the magnitude of the acoustic reflection observedat the center of the antenna.

In order to determine if the forward problem is sensitive to changes in the value ofthe pressure coefficient κ∞, in Figure 16 we plot the absolute value of the differencesin the Ex field component between simulations in which all the parameters exceptκ∞ are fixed at the values given in (69). The solid line in this figure representsthe difference in the Ex field magnitude between the forward simulation in whichκ∞ = 0.8ε∞,0 and the simulation in which κ∞ = 0. The dashed line represents thedifference in the Ex field magnitude between the simulation in which κ∞ = 0.8ε∞,0

and the simulation in which κ∞ = 0.2ε∞,0. We now determine if the forward problemis sensitive to changes in the value of the pressure coefficient κτ . In Figure 17 we plotthe absolute value of the differences in the Ex field component between simulations in

30

0 0.15 0.2 −100

−80

−60

−40

−20

0

20

40

60

80

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

t = 2.5× 10−9 sec

Air

Air−Debye Medium

Pressure waveregion

0.175 0.18

Figure 14: Forward simulation for a Debye medium with parameters given in (69)and (70). Plot of the magnitude of the Ex component of the electric field against z(meters). The reflection from the acoustic pressure wave impinges on the air-Debyeinterface, while the wave transmitted into the pressure region moves into the rightabsorbing layer.

2 2.5 3 3.5 4

x 10−9

−1

−0.5

0

0.5

1

Time t (secs)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

κs = 0

κs = 0.3ε

s,0κ

s = 0.6ε

s,0κ

s = 0.9ε

s,0

Figure 15: Plot of the magnitude of the Ex component of the electric field against z(meters) for different values of the parameter κs. The other parameters are fixed atthe values given in (69) and (70)

31

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−9

0

1

2

3

4

5

6

7x 10−3

Time t (secs)

Abs

olut

e va

lue

of D

iffer

ence

in E

x Fie

ld M

agni

tude

(Vol

ts/m

eter

)

κ∞ = 0.8ε

∞,0 , κ

∞ = 0

κ∞ = 0.8ε

∞,0 , κ

∞ = 0.4ε

∞,0

Figure 16: Plot of the absolute value of the difference in magnitude of the Ex compo-nent of the electric field, between simulations in which κ∞ = 0.8ε∞,0 and simulationswith different values of κ∞, against z (meters). The other parameters are fixed at thevalues given in (69) and (70).

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−9

0

0.005

0.01

0.015

0.02

0.025

Time t (secs)

Abs

olut

e V

alue

of D

iffer

ence

in E

x Fie

ld M

agni

tude

(Vol

ts/m

eter

)

κτ = 0.05τ

0 , κ

τ = 0

κτ = 0.05τ

0 , κ

τ = 0.2τ

0

Figure 17: Plot of the absolute value of the difference in magnitude of the Ex compo-nent of the electric field, between simulations in which the value of κτ = 0.05τ0 andsimulations with different values of κτ , against z (meters). The other parameters arefixed at their true values.

32

which all the parameters except κτ are fixed at the values given in (69). The solid linein this figure represents the difference in the Ex field magnitude between the forwardsimulation in which κτ = 0.05τ0 and the simulation in which κτ = 0. The dashed linerepresents the difference in the Ex field magnitude between the simulation in whichκτ = 0.05τ0 and the simulation in which κτ = 0.2τ0 From these plots we observe thatthe parameter κs seems to be the most influential in the wave interaction, whereas thepressure coefficients κ∞ and κτ do not seem to be as influential. This observation canbe supported from an analysis of (40). In our simulations the outgoing and reflectedradiation will be dominated by frequencies near the center frequency 3.0 GHz. Thus,ε∗r will be dominated by frequencies near 3.0 GHz. In this problem ωτ ∗

0 ≈ O(10−2),and ε0 = 8 × 10−12. Rewriting (40) as

ε∗r =εs

1 + jωτ+

(jωτ

1 + jωτ

)ε∞ +

σ

jωε0

, (74)

we consider each of the three terms in (74) separately to determine their magnitude.The magnitudes of the corresponding true values of these three terms (neglecting theacoustic pressure terms) are approximately given as

(ε∗s,0

1 + jωτ ∗0

)≈ O(ε∗s,0) ≈ O(10),

(jωτ ∗

0

1 + jωτ ∗0

)ε∞,0 ≈ O(10−2ε∗∞,0) ≈ O(10−2),

(σ∗

jωε0

)≈ O(102σ∗) ≈ O(10−3).

(75)

Thus we see that ε∗r will be most sensitive to the static permittivity εs,0 and theeffects of ε∞,0 and τ0 will not be as pronounced. Also, this implies that ε∗r will bemore sensitive to the pressure coefficient κs than to the coefficients κ∞ and κτ , whichcoincides with the observation that was made from Figures 15, 16 and 17

We would also like to see what effect the acoustic speed and frequency have onthe amplitude of the reflection from the acoustic pressure region. In Figure 18 weplot the acoustic reflection observed at the center of the antenna for different valuesof the acoustic frequency and in Figure 19 we plot the acoustic reflection for differentvalues of the acoustic speed. By changing the speed or the frequency, the wavelengthλp changes and thus the size of the interval (z2, z2 +λp) in which the pressure wave isgenerated. The amplitude of the acoustic reflection appears to decrease as the acousticfrequency is increased and increases at the acoustic speed increases. We note herethat our windowed pressure wave contains only one wavelength of the sinusoid. Thusin general it will be difficult to predict precisely how the reflections will behave as thespeed or the frequency is changed.

33

0.17 0.175 0.18 0.185−1

−0.5

0

0.5

1

z (meters)

Win

dow

ed A

cous

tic P

ress

ure

Wav

e

fp = 2.0 × 105 Hz

fp = 3.0 × 105 Hz

fp = 4.0 × 105 Hz

fp = 6.0 × 105 Hz

2 2.5 3 3.5 4

x 10−9

−1

−0.5

0

0.5

1

Time t (secs)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

fp = 2.0 × 105 Hz

fp = 3.0 × 105 Hz

fp = 4.0 × 105 Hz

fp = 6.0 × 105 Hz

Figure 18: Plot of the reflection from the pressure region that is measured at thecenter of the antenna, versus time t (bottom), corresponding to different windowedacoustic pressure waves (top) of varying acoustic frequencies. The other parametersare fixed at the values given in (69) and (70). We note that the amplitude of thereflection changes as the frequency of the pressure wave changes.

34

0.17 0.175 0.18 0.185−1

−0.5

0

0.5

1

z (meters)

Win

dow

ed A

cous

tic P

ress

ure

Wav

e

cp = 1125 m/s

cp = 1500 m/s

cp = 1875 m/s

cp = 2250 m/s

2 2.5 3 3.5 4

x 10−9

−1

−0.5

0

0.5

1

Time t (secs)

Ex F

ield

mag

nitu

de (V

olts

/met

er)

c = 1125 m/sc = 1500 m/sc = 1875 m/sc = 2250 m/s

pppp

Figure 19: Plot of the reflection from the acoustic region that is measured at thecenter of the antenna, against time t (bottom), corresponding to different windowedacoustic pressure waves (top) of varying acoustic speeds. The other parameters arefixed at the values given in (69) and (70). We observe that the magnitude of thereflection is affected by the speed of the acoustic wave.

35

5 Parameter estimation and statistical inference:

The inverse problem

In our forward simulations the signal that we record at the center of the antenna is aset of measurements of Ex, the x component of the electric field, at the point (0, zc)in our computational domain at discrete, uniformly spaced intervals of time. Thissignal has components consisting of the input source, the reflection at the air-Debyeinterface and another reflection from the region that contains the pressure wave asseen in Section 4. The signal is a function of the various dielectric properties of theDebye medium; namely, the static relative permittivity εs,0, the infinite frequencypermittivity ε∞,0, the relaxation time τ0, the conductivity σ, and the three pressurecoefficients κs, κ∞ and κτ . We collect all these quantities together to define theparameter vector

q = [εs,0, ε∞,0, τ0, σ, κs, κ∞, κτ ]T . (76)

Thus, the signal recorded in the forward simulation is a function of q, and changingthe value of q will change the signal that is recorded. In general, such signals areusually obtained in an experimental setting conducted in a laboratory with physicalequipment such as electric pulse generators to generate electric pulses, piezoelectrictransducers that generate the acoustic pressure waves and antennas/receivers thatrecord the electric field intensities. Moreover, such signals generated in an experi-mental setting are usually noisy, with noise arising through the equipment that isused; namely the resistor, antenna/receiver and the transducer, in our case. Implicitin such collection of measurements is the assumption that there exist true values ofall the parameters that characterize the medium to be interrogated. We will denotethe corresponding true parameter vector by q∗.

In order to simulate an experimental situation (i.e., generate typical “data”) weassume that there exist true values of all the parameters involved in our problem,and we use these true values (which of course would not be known in an experimentalsetting!) in our forward simulation, viz. the FDTD method and record the signalobserved at the antenna as described above. We then add noise to the generated signalto produce a noisy signal which will form the observations or data that we will use in aninverse problem, as a substitute for data generated in the laboratory. The goal of ourelectromagnetic interrogation technique is to identify or estimate the true parameterq∗ that characterizes the particular Debye medium being interrogated, from datathat consists of a signal recorded at the center of the antenna. The reason for usingsimulated data in the inverse problem is to validate the methods first on “data” fromknown parameters in a setting with known noise. If we cannot estimate the dielectricparameters from data that is constructed via numerical simulations of our discretemodel, then the reconstruction of these parameters from actual experimental datausually will not be feasible.

We can state the inverse or parameter estimation problem that we will attempt to

36

solve as follows: Using observations or data (electric field intensities containing noisecollected at the center of the antenna), determine an estimate q, of the true parameterq∗, that belongs to an admissible set Q, so that the solution of the forward problemwith q = q best describes the data that is collected. When we have solved thisdeterministic problem (as will be detailed below), it will also be necessary to specifyreliablility of our estimates, i.e., can we specify measures of uncertainty related tothe estimate q of q∗? It is important to note that the uncertainty in consideration isinherent in the method of producing the estimates as well as in the process of datacollection. Such measures of uncertainty will be specified by means of confidenceintervals. These intervals are a probability statement about the procedure whichis used to construct estimates of the parameters. Thus we are led in a completelynatural way to stochastic or probabilistic aspects of estimates from a deterministicproblem solved with deterministic algorithms [BB99]. The statistical error analysisthat we will present here is based on standard statistical formulations as given in[DG95].

We first consider the two different ways in which we can add noise to our signal,i.e., the values of the x component, Ex, of the electric field observed at the center ofthe antenna, (0, zc), for different times tk = k∆t, k = 0, 1, . . . ,M . We represent thissignal as the vector

(i) E(q∗) = E∗kM

k=1 = (Ex(t1, 0, zc;q∗), . . . , Ex(tM , 0, zc;q

∗))T ,

(ii) q∗ = (ε∗s,0, ε∗∞,0, τ

∗0 , σ∗, κ∗

s, κ∗∞, κ∗

τ )T ,

(77)

q∗ being the true parameter vector.

1. Relative random noise (RN) : In this case the amplitude of the noise that isadded is proportional to the size of the signal, E∗

kMk=1. Our simulated data is

Osk = E∗

k(1 + νηsk), k = 1, . . . M, (78)

where η = ηkMk=1 are independent normally distributed random variables with

mean zero and variance one, i.e., ηk ∼ N (0, 1), k = 1, . . . ,M . We express therelative magnitude of the noise as a percentage of the magnitude of E(q∗) bytaking two times the value of ν as the size of the random variable. For example,ν = 0.005 corresponds to 1% relative noise [BBL00]. This noise model does notproduce constant variance across the samples.

2. Constant variance noise: Since constant variance is most conveniently assumedin standard error analysis, we will consider estimates obtained from an inverseproblem applied to simulated data which contains constant variance randomnoise. The data that we try to fit in this case is

Osk = E∗

k + βηsk, k = 1, . . . M, (79)

37

where as in (78), ηk ∼ N (0, 1), k = 1, . . . ,M . The constant β is taken to bethe product να, where α is a scaling factor which is chosen so that the signalto noise ratio (SNR) of the data defined in (79) corresponds to data defined in(78) with noise level ν. This justifies comparing results obtained with the twodifferent ways of adding comparable noise to our signal. This will be addressedin more detail in Section 5.

The vector Os = OskM

k=1 will be our data or observations, and the noise η = ηkMk=1

will be referred to as the measurement errors corresponding to the observations.The statistical error analysis that we now develop is only applicable to the case ofconstant variance noise. Thus, for the rest of this section, we will assume that ourobservations are of the form (79). The sample observations Os, which are in generalobtained from experiment, are a realization of the corresponding random variableO = (O1, O2, . . . , OM)T which can be seen to be a transformation of the randomvariable η. Thus

Ok = E∗k + βηk, k = 1, . . . M, (80)

is a stochastic process and has a multivariate normal distribution with mean vectorE(q∗) defined in (77, i) and covariance matrix β2IM . The statistical model

O ∼ NME(q∗), β2IM, (81)

is a formal representation of the population of all possible realizations of O1, O2, . . . , OM

that can be observed. When we collect data, we are observing a single realizationof Os

1, . . . , OsM i.e., a sample. Our objective then is to estimate the true value of the

parameter q∗ by collecting data, i.e., observing a single realization of O as well asaccounting for the fact that a different realization will produce a different estimate ofq∗. We would like to learn about the true value q∗ (which determines the nature ofthe population) from a sample, as well as indicate the certainty (or uncertainty) thatwe can associate to our knowledge of this parameter. This process of making state-ments about a population of interest on the basis of a sample from the population iscalled statistical inference.

The unknown true parameters vector q∗ will be estimated by means of a suit-able function q(O) of the observations O. The function q(O) is a random vari-able and is referred to as an estimator. When evaluated at a particular realization,Os

1, Os2, . . . , O

sM , this estimator yields a numerical value that gives information about

the true value of the parameter q∗. For a particular realization Os of the randomvariable of observations O, we call q(Os) an estimate. In this report we will considerthe method of least squares to obtain estimates for the parameters and to calculateconfidence intervals for the estimates by linearizing around the estimate q(Os). Inthis case q(O) will be the least squares estimator qOLS(O). Associated with the ran-dom variable qOLS(O) is the probability space Q = (Q,B,m), where Q is the sampleset of all admissible parameters q, B is the σ-algebra of events and m is the probabil-ity measure (or distribution). Thus we look for the least squares estimator qOLS(O)

38

on Q such that

qOLS(O) = arg minq on Q

J(q),

J(q) =1

2

M∑

k=1

|Ex(tk, 0, zc; q) − Ok|2 ,(82)

where E(q) = (Ex(t1, 0, zc;q), Ex(t2, 0, zc;q), . . . , Ex(tM , 0, zc;q))T will be generatedby our forward simulation. We will refer to the vector E(q) as our simulations. For aparticular realization Os of the random variable O of observations, this process willyield the least squares estimate qOLS(O

s) ∈ Q, where

qOLS(Os) = arg min

q∈QJS(q),

JS(q) =1

2

M∑

k=1

|Ex(tk, 0, zc;q) − Osk|2 ,

(83)

5.1 Calculation of confidence intervals via linearization

In general, the mappings between the parameters q and the simulations E are non-linear. Let Ek = Ex(tk, 0, zc;q). The functions

E1(q1, . . . , ql) = Ex(t1, 0, zc;q),

E2(q1, . . . , ql) = Ex(t2, 0, zc;q),

...EM(q1, . . . , ql) = Ex(tM , 0, zc;q),

(84)

denote real-valued differentiable functions of the unknown parameters q = (q1, . . . , ql)T ,

where 1 ≤ l ≤ 7 depending on how many parameters we attempt to identify. Letq = q∗ +Dq, where the corrector term Dq = (∆q1, . . . ∆ql)

T is unknown. Linearizingthe functions in (84) around the true parameter q∗ by using the Taylor expansion weobtain

Ek(q1, . . . ql) = Ek(q∗1 + ∆q1, . . . , q

∗l + ∆ql)

= Ek(q∗) +

l∑

j=1

∂Ek

∂qj

q∗

∆qj, k = 1 . . . M.(85)

Let us defineDe = (O1 − E1(q

∗), . . . , OM − EM(q∗))T , (86)

and

X (q∗) =

∂E1

∂q1

q∗

. . . ∂E1

∂ql

q∗

......

...∂EM

∂q1

q∗

. . . ∂EM

∂ql

q∗

. (87)

39

Thus, in order to estimate the unknown corrector terms by the method of leastsquares, the estimator DOLS is determined by

DOLS = minDq

1

2(De −X (q∗)Dq)

T (De −X (q∗)Dq), (88)

which is the linearized version of (82). This involves solving

∂(DT

e De − 2DTe X (q∗)DOLS + DT

OLSX (q∗)TX (q∗)DOLS

)

∂DOLS

= 0, (89)

and the corresponding optimality condition yields the estimator

DOLS = (X (q∗)TX (q∗))−1X (q∗)TDe. (90)

We note that the expected value of DOLS is Exp(DOLS) = 0, and the covariance matrixof DOLS and hence also of qOLS is

Cov(DOLS) = (X (q∗)TX (q∗))−1X (q∗)T Cov(De)X (q∗)(X (q∗)TX (q∗))−1

= β2(X (q∗)TX (q∗))−1 = Cov(qOLS),(91)

where qOLS = q∗ + DOLS, and β is defined in (79). Thus the least squares estimatoris a random variable with a multivariate normal distribution which after linearizationcan be approximately represented as

qOLS(O) ∼ Nl

(q∗, β2[X T (q∗)X (q∗)]−1

). (92)

However, when data is generated in an experimental setting, we do not have any

knowledge of the true parameter vector q∗ and hence we cannot calculate(X (q∗)TX (q∗)

)−1.

Thus, we further approximate our least squares estimator as having the multivariatenormal distribution

qOLS(O) ∼ Nl

(qOLS(O

s), β2OLS[X T (qOLS(O

s))X (qOLS(Os))]−1

), (93)

which is obtained by repeating the linearization process around the estimate qOLS(Os)

obtained from a realization Os of the random variable O of observations. The valueof β2

OLS is chosen as

β2OLS =

1

M − l

M∑

k=1

|Ex(tk, 0, zc; qOLS(Os)) − Os

k|2 , (94)

which is the minimum value of the least squares objective function scaled by 2/(M−l).Whenever an estimate based on data is reported, it should be accompanied by

an assessment of uncertainty based on the sampling distribution. To this end we

40

will construct confidence intervals for all the estimates that we will provide. Theapproach we use is to look at the standard error (SE) for each of the l componentsof the estimate qOLS(O

s), which is given for its jth component by the jth diagonalterm of the covariance matrix, i.e.,

SE(qOLS,j) =√

var(qOLS,j(Os))

=√

β2OLS ((X (qOLS(Os))TX (qOLS(Os)))−1)jj j = 1, . . . , l.

(95)

We construct the intervals

CIj = (qOLS,j(Os) − 1.96SE(qOLS,j), qOLS,j(O

s) + 1.96SE(qOLS,j)) , j = 1, . . . , l,(96)

for which

m (qOLS,j(Os) − 1.96SE(qOLS,j) < qOLS,j(O) < qOLS,j(O

s) + 1.96SE(qOLS,j)) = 0.95,

j = 1, . . . , l,

(97)

where m is the probability measure for the probability space Q, and M is sufficientlylarge that one can use a Gaussian N (0, 1) distribution in computing confidence in-tervals. These intervals CIj are called confidence intervals with confidence level 0.95or the 95% confidence interval.

Remark 2 The confidence intervals are a probability statement about the procedureby which an estimate is constructed from a sample of the population. They are not aprobability statement about the true parameter q∗ which is fixed. If we could constructthe confidence intervals, from our estimation procedure, for all possible data samplesof size M, then 95% of such intervals would cover the true parameter values q∗.However, for a particular data sample Os, the confidence intervals constructed asabove may or may not cover the true parameter q∗. What we can state is that we are95% confident that the confidence intervals constructed by our estimation procedurewould cover q∗.

6 Parameter estimation: First test problem

We now attempt to estimate the dielectric parameters for the Debye medium definedin (69) and (70). We restate the values of all the parameters below

ε∗s,0 = 80.1

ε∗∞,0 = 5.5

τ ∗0 = 8.1 × 10−12

σ∗ = 1 × 10−5

κ∗s = 48.06

κ∗∞ = 4.4

κ∗τ = 4.05 × 10−13.

(98)

41

We will refer to the values (98) as the true values of the parameters. We will solvethe corresponding least squares optimization problem using two different methods;the Nelder-Mead algorithm, which is a simplex based, gradient free method, andthe Levenberg-Marquardt trust region method that uses forward finite differences tocalculate the gradient. From our estimates for all the seven parameters we will showthat our problem is sensitive to only three of the seven parameters. We will thentry to estimate two or three of the parameters to which our problem is sensitive.The particular implementation of the Levenberg-Marquardt and the Nelder-Meadalgorithm that are used in this report are based on the corresponding algorithmspresented in [Kel99]

6.1 Simulation results: The Nelder-Mead method

We first present parameter estimation results using the Nelder-Mead simplex basedalgorithm which is used for optimization in noisy problems. The Nelder-Mead simplexalgorithm keeps a simplex S of approximations to an optimal point. In this algorithmthe vertices of the simplex qil+1

i=1, where l is the size of q, are sorted according tothe least squares objective function values

JS(q1) ≤ JS(q2) ≤ . . . JS(ql+1). (99)

q1 is called the best vertex and ql+1 the worst. The algorithm attempts to replacethe worst vertex ql+1 with a new point of the form

q(νnm) = (1 + νnm)q − νnmql+1, (100)

where q is the centroid of the convex hull of qil+1i=1

q =1

l

l∑

i=1

qi. (101)

The value of νnm is selected from a sequence of values that are computed by performingseveral different operations on the simplex. In general, the Nelder-Mead algorithm isnot guaranteed to converge, even for smooth problems. The failure mode is stagnationat a nonoptimal point. For further details we refer the reader to [Kel99].

In the first test we will try to estimate all of seven parameters in q∗. The initialsimplex is chosen to have values that are 5-10% larger than the true parameter valuesgiven in (98). We do not add any noise to our data that is used in the parameterestimation in this first test. The final estimates from the Nelder-Mead algorithmare presented in Table 1. The details of the simulation are plotted in Figures 21-22. Figure 21 plots the various features of the Nelder-Mead run as functions of thenumber of iterations. The algorithm is terminated when the maximum least squaresfunction value difference between any two points in the simplex is less than 10−8.

42

Parameter True Values Final Estimatesεs,0 80.1 80.1111ε∞,0 5.5 5.7461τ0 8.1 × 10−12 8.1283 × 10−12

σ 1.0 × 10−5 1.1972 × 10−5

κs 48.06 48.1227κ∞ 4.4 4.9148κτ 4.05 × 10−13 4.1935 × 10−13

Table 1: Final estimates for all seven param-eters for water using Nelder-Mead. The ini-tial parameter simplex has values that are5-10% larger than the true values.

Function Difference 7.6979 × 10−9

Max Oriented Length 2.4986 × 10−3

Average least squares value 9.8451 × 10−6

Best Least squares value 9.8392 × 10−6

L2 norm of the gradient 1.008 × 10−2

L∞ norm of gradient 8.756 × 10−3

Number of forward solves 453

Table 2: Status of the Nelder-Mead Simula-tion at the 283rd iteration.

43

0 100 200 30078

80

82

84

86

Iterations

ε s,0

^

εs,0* = 80.1

0 100 200 30046

48

50

52

Iterations

κ s^

κs* = 48.06 (ε

s)

0 100 200 3008.1

8.3

8.5

8.7

8.9

x 10−12

Iterations

τ

τ

0^

0* = 8.1e−12

Figure 20: Variation of εs,0, τ0 and κs with respect to iteration number. Out of allseven parameters these three parameters are relatively well estimated.

The algorithm converges in 283 iterations. Table 2 presents the final status of thesimulation. In Figure 21, the maximum oriented length is defined as

Max Oriented length = max2≤i≤l+1

||q1 − qi||. (102)

Also the gradient in this case refers to the simplex gradient. As can be seen fromFigure 22 the parameters ε∞,0, σ and the two pressure coefficients κ∞ of ε∞,0 andκτ of τ0 are difficult for the inverse problem to identify. On the other hand, fromFigure 20 we see that the parameters εs,0 and its pressure coefficient κs as well asthe parameter τ0 are relatively well estimated. Since the inverse problem is only ableto identify three parameters in the absence of noise we do not expect to see anyimprovement when noise is added to the data! Thus we will henceforth concentrateon the identification of εs,0, τ0 and κs.

We will first attempt the identification of εs,0 and κs. In a first test, we fix allthe other 5 parameters (including τ0) at the true values and attempt to identify εs,0

and κs. Table 3 presents the results for this test. As can be seen from Table 3, withthe other 5 parameters fixed at their true values we do not have any difficulty inidentifying εs,0 and κs accurately. In Table 3, FC denotes the function count, i.e., thenumber of times the least squares objective function is evaluated. In Figure 23 weplot the values of εs,0 (left) and κs (right) over all iterations, for the three differentinitial simplicies with values that are 5%-10% lower than q∗ (o-), 5% -10% higherthan q∗(- -), and 50%-60% lower than q∗(-).

In Figure 24 we plot the details of the Nelder-Mead simulations for the threedifferent initial simplices tabulated in Table 3. In each case the algorithm convergeswhen the difference between function values is less than 10−8. For the second case

44

0 50 100 150 200 250 30010−4

10−2

100

102

104

0 50 100 150 200 250 30010−10

10−5

100

105Function Differences

0 50 100 150 200 250 30010−6

10−4

10−2

100

102

Least Squares Function Value

0 50 100 150 200 250 30010−4

10−3

10−2

10−1Max Oriented Length

0 50 100 150 200 250 30010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 50 100 150 200 250 30010−6

10−4

10−2

100

102Average Least Squares value

L∞ Norm of Gradient

Iterations

Figure 21: Nelder-Mead for the estimation of seven parameters for water.

45

0 100 200 3005.7

5.8

5.9

6

Iterations

ε ∞,0

^

ε∞,0* = 5.5

0 100 200 3001.04

1.08

1.12

1.16

1.2x 10−5

Iterations

σ

σ* = 1.0e−5

0 100 200 3004.4

4.6

4.8

5

Iterations

κ ∞^

κ∞* = 4.4

0 50 100 150 200 250 300

4.2

4.25

4.3

4.35

x 10−13

Iterations

κ τ^

κτ* = 4.05e−13

Figure 22: Variation of ε∞,0, σ and estimates of the two pressure coefficients, namely,κ∞ and κτ , with respect to iteration number. As can be seen our inverse problem isnot sensitive to these four parameters and their identification is difficult even in theabsence of noise.

46

0 10 20 30 40 50 60

50

60

70

80

Iterations

ε s,0

^

εs,0* = 80.1

0 10 20 30 40 50 600

10

20

30

40

50

60

Iterations

κ s^

κs* = 48.06

Figure 23: Plots of εs,0 (left) and κs (right) against the number of iterations forsimulations in which the other five parameters are fixed at their true values and nonoise is added to the data. (o-) 0.95q∗, (- -) 1.05q∗, (-) 0.5q∗.

Table 3: Parameter estimation of εs,0 and κs for different initial simplices with noadded noise.

q0 Iter εs,0 κs |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.95q∗ 33 80.0999 48.0607 4.7019×10−9 1.4560×10−8 0.0204 651.05q∗ 32 80.1 48.06 9.5079×10−9 7.1205×10−27 0.0096 660.5q∗ 59 80.1001 48.0603 8.4746×10−9 5.5171×10−9 0.0130 112

47

where the initial simplex has points with 5%-10% larger values than the true values,we obtain an unusually small least squares function value of the order of 10−27. Webelieve this to be an accidental artifact of the algorithm. In Figure 25 and 26 we plotthe least squares objective function for values of εs,0 and κs that differ by 0%-10% fromthe true values. The other five parameters are kept fixed at their true values. FromFigures 25 and 26, we determine that our inverse problem is more sensitive to εs,0 thanto the pressure coefficient κs. This is expected as the static relative permittivity, εs,0

is more influential in the system dynamics than the pressure coefficients, and henceis also more important in identifying and characterizing the material.

As explained in Section 5, in a more realistic situation we will not have accessto the true values of any of the parameters. Hence, we now fix five of the sevenparameters at values that have a relative error from the true values of about 10%-50%. We use the values

ε∞,0 = 6.0,τ0 = 10.0 × 10−12,σ = 1.5 × 10−5,κ∞ = 4.8,κτ = 5.0 × 10−13,

(103)

in our simulations. Such choices for the parameters are typical in testing algorithms[BK89]. We use different initial guesses for εs,0 and κs. Also in an experimental settingnoise is introduced into the data, as discussed earlier. To simulate the experimentalprocess in data collection we first add relative random noise to our data, and we usethe fixed values (103) in the inverse problem. The results are tabulated in 4. FromTable 4 we see that using the inverse problem we are able to estimate εs,0 but weare unable to reconstruct κs with good accuracy. In Figure 27 we plot the values ofεs,0 (left) and κs (right) over all iterations, In Figure 28 we plot the details of theNelder-Mead simulation for the results in Figure 27 and Table 4 which show the finalestimates of εs,0 and κs for water. The other five parameters are fixed at the valuesgiven in (103) and relative noise of varying levels is added to the data.

To understand why κs is not recovered, we note that the fixed value of τ0 in (103)has a relative error of 23% from its true value. Since the reflections from the acoustic

Table 4: Parameter estimation of εs,0 and κs for 0%, 1%, 3% and 5%relative noise.

% RN Iter εs,0 κs |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.0 37 80.3636 66.0486 8.5719×10−9 2.5594 0.0076 731.0 41 80.3638 66.0422 3.2939×10−9 25.2195 0.0098 803.0 40 80.3640 66.0280 8.6023×10−9 208.1146 0.0080 775.0 40 80.3643 66.0157 6.3563×10−9 574.4429 0.0060 79

48

0 10 20 30 40 50 6010−4

10−2

100

102

104L∞ Norm of Gradient

0 10 20 30 40 50 6010−10

10−5

100

105Function Differences

0 10 20 30 40 50 6010−30

10−20

10−10

100

1010

Iterations

Least Squares Function Value

0 10 20 30 40 50 6010−5

100Max Oriented Length

0 10 20 30 40 50 6010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 10 20 30 40 50 6010−10

10−5

100

105Average Least Squares value

Figure 24: Nelder-Mead for the estimation of εs,0 and κs for water. The other fiveparameters are fixed at their true values and no noise is added to the data. (o-)0.95q∗, (- -) 1.05q∗, (-) 0.5q∗.

49

JS

7075

8085

90

4244

4648

5052

540

20

40

60

80

100

εs,0

Least Squares Objective Function

κs

Figure 25: The Least Squares Objective Function for different values of εs,0 and κs.We can see from this plot that the problem is more sensitive to εs,0 than to thepressure coefficient κs.

44 46 48 50 520

0.1

0.2

0.3

0.4

κs

72 76 80 84 880

20

40

60

80

εs,0

JJ SS

Least Squares Objective Function Values

Figure 26: Cross-sections of the Least Squares Objective function versus κs (left) andversus εs,0 (right). Note the difference in scales between the two figures.

50

0 10 20 30 40 5072

74

76

78

80

82

Iterations

ε s,0

^

εs,0* = 80.1

0 10 20 30 40 5035

40

45

50

55

60

65

70

Iterations κ s^

κs* = 48.06

Figure 27: Plots of εs,0 (left) and κs (right) against the number of iterations forsimulations in which the other five parameters are fixed at values that have 10%-50%relative error from the true values (103) and relative noise of varying levels is addedto the data.

region depend strongly on the value of τ0, a relative error of 23% is too high to geta good estimate of the pressure coefficients. To test this we lower the value of τ0 to8.91 × 10−12. So instead of the values in (103) we will use the values

ε∞,0 = 6.0,τ0 = 8.91 × 10−12,σ = 1.5 × 10−5,κ∞ = 4.8,κτ = 5.0 × 10−13.

(104)

This value of τ0 has a relative error of 10% from its true value τ ∗0 . With this new

fixed value for τ0 we attempt the inverse problem again to reconstruct εs,0 and κs.These results are presented in Table 5. From Table 5 we observe that the estimatedvalue of κs is closer to the true value than in the previous test. Thus we can obtainaccurate estimates of κs by choosing values of τ0 that are closer to its true value. Wecan conclude that if the value of τ0 is not accurately known, then the identificationof κs is difficult. The identification of εs,0 on the other hand does not appear to besensitive to the fixed value of τ0 that is chosen. The combined identification of bothεs,0 and κs appears to be quite insensitive to the level of relative noise that is addedto the data. The results of the Nelder-Mead run and the variation of the parametersover all iterations are plotted in Figures 29 and 30, respectively.

51

0 10 20 30 40 5010−4

10−2

100

102

104L∞ Norm of Gradient

Iterations

0 10 20 30 40 5010−10

10−5

100

105Function Differences

0 10 20 30 40 50100

101

102

103Least Squares Function Value

0 10 20 30 40 5010−5

100Max Oriented Length

0 5 10 15 20 25 30 35 4010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 5 10 15 20 25 30 35 40100

101

102

103Average Least Squares value

Figure 28: Nelder-Mead for the estimation of εs,0 and κs for water. The other fiveparameters are fixed at the values given in (103) and relative noise of varying levelsis added to the data. 0% (–), 1% (- -) 3% (o-) and 5% (+-)

52

Table 5: Parameter estimation of εs,0 and κs for 0% - 10% added relativerandom noise. The values of the other five parameters are fixed at valuesgiven in (104).

% RN Iter εs,0 κs |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.0 38 80.2165 55.2945 1.5487×10−9 0.4369 0.0013 730.1 38 80.2165 55.2949 7.2106×10−9 0.6551 0.006310 750.3 39 80.2166 55.2946 5.4008×10−9 2.4674 0.005495 760.5 39 80.2167 55.2939 1.8372×10−9 6.1140 0.009746 761.0 39 80.2169 55.2914 4.5901×10−9 23.2556 0.0075 763.0 41 80.2173 55.2826 4.2341×10−9 206.4680 0.0087 785.0 40 80.2178 55.2729 8.2103×10−9 573.1135 0.0079 7810.0 39 80.2191 55.2509 2.5361×10−9 2292.24730 0.00701 76

0 10 20 30 40 5072

74

76

78

80

82

Iterations

ε s,0

^

εs,0* = 80.1

0 10 20 30 40 5038

42

46

50

54

58

Iterations

κ s^

κs* = 48.06

Figure 29: Plots of εs,0 (left) and κs (right) against the number of iterations forsimulations in which the other five parameters are fixed at values that have 10%-50%relative error from the true values as given in (104), and relative noise of varyinglevels is added to the data.

53

0 10 20 30 40 5010−4

10−2

100

102

104L∞ Norm of Gradient

0 10 20 30 40 5010−10

10−5

100

105Function Differences

0 10 20 30 40 5010−2

100

102

104Least Squares Function Value

0 10 20 30 40 5010−5

100Max Oriented Length

0 10 20 30 40 5010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 10 20 30 40 5010−2

100

102

104Average Least Squares value

Iterations

Figure 30: Nelder-Mead for the estimation of εs,0 and κs for water. The other fiveparameters are fixed at the values in (104), and relative random noise of varying levelsis added to the data.

54

We next attempt the inverse problem with simulated data to which we add con-stant variance noise. We would like to compare the results of this case with the resultsobtained from the inverse problem by adding relative random noise to the simulateddata. We use the notion of signal to noise ratio to make such comparisions.

6.2 Signal to noise ratio

In analog and digital communications, the signal to noise ratio, denoted as SNR,is a measure of the signal strength relative to the background noise. The ratio isusually measured in decibels (dB). There are many equivalent ways of defining theSNR [JN84]. Since we assume the mean of our noise to be zero, we define the SNRas

SNR = 10 log10

( ∑Mk=1 |E∗

k |2∑Mk=1 var(Ok − E∗

k)

)dB, (105)

where dB denotes decibels. If the signal strength is equal to the variance of thebackground noise, i.e.,

M∑

k=1

|E∗k |2 =

M∑

k=1

var(Ok − E∗k) (106)

then SNR = 0, and the signal is almost unreadable, as the noise level severely com-petes with it. Ideally, the strength of the signal is greater than the variance of thenoise, so that the SNR is positive which results in the signal being readable. If thestrength of the signal is less than the variance of the noise, then the SNR is negative.Communications engineers always strive to maximize the SNR ratio. The SNR ratiocan be increased by providing the source with a higher level of signal output powerif necessary. In wireless systems, it is always important to optimize the performanceof the transmitting and receiving antennas.

Using the notation presented in Section 5, the SNR in the cases of relative randomnoise SNRR and constant variance noise SNRC are, respectively, given as

SNRR = 10 log10

( ∑Mk=1 |E∗

k |2∑Mk=1 var(νE∗

kηk)

)dB,

SNRC = 10 log10

( ∑Mk=1 |E∗

k |2∑Mk=1 var(βηk)

)dB

(107)

To compare the case of constant variance noise with the case of relative random noise,we proceed as follows.

1. As noted in (79) and the discussion thereafter, we set β = να. Setting the SNR

55

Table 6: Comparison of relative ran-dom noise and constant variance noise

SNR (dB) ν % RN β = να66 0.0005 0.1 0.011856 0.0015 0.3 0.035352 0.0025 0.5 0.058846 0.005 1.0 0.117536 0.015 3.0 0.352632 0.025 5.0 0.587726 0.05 10.0 1.1754

in the two cases to be equal i.e., SNRR = SNRC, implies

M∑

k=1

var(νE∗kηk) = Mν2α2 =⇒ α =

√∑Mk=1 |E∗

k |2M

. (108)

2. Thus for a given value of ν, we have β = αν and the corresponding SNR iscalculated as

SNRR ≡ SNRC = 10 log10

(∑Mk=1 |E∗

k |2Mα2ν2

)= 10 log10

(1

ν2

). (109)

Hence, for each value of ν there is a corresponding value of β for which the SNR forrelative random noise is equivalent to the SNR for constant variance noise. In Table6 we present the values of the SNR, β, and ν that correspond to each other in themanner described above. For this test problem α = 23.5077.

We next attempt to identify the two parameters εs,0 and κs by adding constantvariance noise as given in (79) to our data and using the fixed values (104). Theresults for the corresponding inverse problem are presented in Table 7. The results inthis case are quite different from the case of relative noise as shown in Table 5. Wecan see that the estimation of κs becomes worse as the level of noise increases. Theestimation of εs,0 on the other hand is relatively stable, though it is not as accurateas the estimate obtained in the case where relative noise is added to the data. Theestimates also become worse as the level of constant variance noise increases. In Table7, ν corresponds to the scaling factor in (78). The constant variance case and therelative noise case for the same value of ν have the same signal to noise ratio. Figure31 plots the variation of εs,0 and κs over all iterations. As can be seen here the valueof κs increases, away from the true value, as the noise level increases. Figure 32 plotsthe details of the corresponding Nelder-Mead Simulation.

56

0 5 10 15 20 25 30 35 40 4572

73

74

75

76

77

78

79

80

81

Iterations

ε s,0

^

εs,0* = 80.1

0 5 10 15 20 25 30 35 40 4535

40

45

50

55

60

65

70

75

Iterations

κ s^

κs* = 48.06

Figure 31: Plots of εs,0 (left) and κs (right) against the number of iterations forsimulations in which the other five parameters are fixed at values that have 10%-50% relative error from the true values as given in (104), and constant variance noisecorresponding to varying levels of relative noise is added to the data.

Table 7: Parameter estimation of εs,0 and κs for 0%-10% constant variance noise. Theother five parameters are fixed at the values given in (104)

% RN Iter εs,0 κs |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.0 38 80.2165 55.2945 1.5487×10−9 0.4369 0.0013 730.1 38 80.2114 55.4112 8.1590 × 10−9 0.6973 0.01237 740.3 41 80.2012 55.6422 9.6609 × 10−9 2.8205 0.001064 770.5 40 80.1910 55.8728 8.1572 × 10−9 7.0804 0.0005887 771.0 41 80.1654 56.4531 3.1244 × 10−9 27.0778 0.007561 823.0 40 80.0636 58.7953 9.7228 × 10−9 240.6053 0.01135 775.0 39 79.9621 61.1779 7.8005 × 10−9 667.7927 0.0111 7910.0 44 79.7084 67.3235 3.5288 × 10−10 2670.5115 0.007417 85

57

Table 8: Confidence Intervals for the parameter estimation of εs,0 and κs for constantvariance noise with the Nelder-Mead algorithm.

RN = 0.0%εs,0 (8.02165 ± 1.8804 × 10−3) × 10κs (5.52945 ± 1.6448 × 10−2) × 10

RN = 0.1%εs,0 (8.02114 ± 2.3718 × 10−3) × 10κs (5.54112 ± 2.0745 × 10−2) × 10

RN = 0.3%εs,0 (8.02012 ± 4.7554 × 10−3) × 10κs (5.56422 ± 4.1583 × 10−2) × 10

RN = 0.5%εs,0 (8.01910 ± 7.5113 × 10−3) × 10κs (5.58728 ± 6.5668 × 10−2) × 10

RN = 1.0%εs,0 (8.01654 ± 1.4577 × 10−2) × 10κs (5.64531 ± 1.2738 × 10−1) × 10

RN = 3.0%εs,0 (8.00636 ± 4.2160 × 10−2) × 10κs (5.87953 ± 3.6783 × 10−1) × 10

RN = 5.0%εs,0 (7.99621 ± 6.8206 × 10−2) × 10κs (6.11779 ± 5.9447 × 10−1) × 10

RN = 10.0%εs,0 (7.97084 ± 1.2723 × 10−1) × 10κs (6.73235 ± 1.1076) × 10

We now calculate confidence intervals for the estimates that we obtained in Table7. The procedure for calculating these intervals was outlined in Section 4.1. Theseintervals are presented in Table 8 for varying levels of noise. From our earlier dis-cussion about confidence intervals, we note that the size of the intervals is in directproportion to the level of uncertainty that we have about the estimates that we haveobtained. Thus, the smaller the confidence interval the more confident we are aboutthe estimates obtained. As we can see from Table 8 we have tighter intervals for theestimates of εs,0 than we do for the estimates of κs, which is expected as our problemis more sensitive to εs,0 than it is to the pressure coefficient κs. We also notice thatthe intervals become larger as the level of noise that is added to the data is increased.These intervals correspond to a 95% confidence level.

58

0 10 20 30 40 5010−4

10−2

100

102

104L∞ Norm of Gradient

0 10 20 30 40 5010−10

10−5

100

105Function Differences

0 10 20 30 40 5010−2

100

102

104Least Squares Function Value

0 10 20 30 40 5010−5

100Max Oriented Length

0 10 20 30 40 5010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 10 20 30 40 5010−2

100

102

104Average Least Squares value

Iterations

Figure 32: Nelder-Mead for the estimation of εs,0 and κs for water. The other fiveparameters are fixed at the values given in (104), and constant variance noise corre-sponding to varying levels of relative noise is added to the data.

59

JS

7678

8082

84

3540

4550

556025

30

35

40

45

50

55

εs,0

Least Squares Objective Function (ν = 0.005)

κs

78 80 82 84

30

35

40

45

εs,0

JS

40 45 50 55

27

27.5

28

28.5

κs

JS

Least Squares Objective Function Cross−sections

Figure 33: (Top) The Least Squares Objective Function for different values of εs,0

and κs. Constant variance noise with ν = 0.005 (1% RN) is added to the data. Theother five parameters were fixed at their true values. The minimum for this plot islocated at the point (80.099, 49.0212), with a minimum value of 26.712. (Bottom)Cross sections of the Least Squares Objective function versus κs (left) and versus εs,0

(right).

60

JS

7678

8082

84

35

40

45

50

55

6025

30

35

40

45

50

εs,0

The Least Squares Objective Function (ν = 0.005)

κs

78 80 82 84

30

35

40

45

εs,0

JS

40 45 50 55

28

29

30

31

κs

JS

Least Squares Function Cross−Sections

Figure 34: (Top) The Least Squares Objective Function for different values of εs,0 andκs. Constant variance noise with ν = 0.005 (1% RN) is added to the data. The otherfive parameters were fixed at the values given in (104). The minimum is located at thepoint (80.099, 56.7108), with a minimum value of 27.0837. (Bottom) Cross sectionsof the Least Squares Objective function versus κs (left) and versus εs,0 (right).

61

0 1 2 3 40

5

10

15

Time (sec)

Res

idua

l (V

olts

/Met

er)

Figure 35: Plot of the Least Squares residual versus time in the case of 10% relativerandom noise added to the data.

In Figure 33 we plot the least squares objective function for values of εs,0 thathave up to 5% relative error from the true values, and values of κs that have up to20% relative error from the true values. The other five parameters were fixed at theirtrue values in the case of the top plot in Figure 33 and at the values (104) in the caseof the bottom plot, respectively. In Figure 34 the top and bottom plots show thethe cross sections of the objective function against εs,0, and κs across the minimumwith respect to the other variable, for the surface plots in Figure 33. Comparing withFigure 25 we observe that the minimum value of κs has moved away from its truevalue.

In Figures 35 we plot the least squares residual versus time in the case when 10%relative random noise is added to the data. A magnified plot of the least squaresresidual versus time in the interval where the acoustic reflection is observed is plottedin Figure 36. Finally we plot the least squares residual versus the absolute valueof the electric field data in Figure 37 again in the case where 10% relative randomnoise is added to the data. We notice in this figure that the amount of noise increasesas the absolute value of the electric field data magnitude increases. This is in directcontrast to the case when constant variance noise is added to the simulated data. Weplot analogous plots of the least squares residual for 10% added constant variancenoise in Figures 38 and 39. Comparing Figures 38 and 35 we note the lack of anypatterns in the residual in Figure 38 for the case of added constant variance noise;whereas in Figure 35, the residual follows the peaks in the simulated data used in

62

2.5 3 3.5 4

x 10−9

0

0.1

0.2

Time (sec)

Res

idua

l (V

olts

/Met

er)

Figure 36: A magnified plot of the Least Squares residual versus time in the intervalwhere the acoustic reflection is observed, for the case of 10% relative random noiseadded to the data.

0 50 100 1500

5

10

15

Absolute Value of the Electric Field, |Ex|

Res

idua

l (V

olts

/Met

er)

Figure 37: Plot of the Least Squares residual versus the absolute value of the electricfield data for the case of 10% relative random noise added to the data.

63

0 1 2 3 4

x 10−9

0

1

2

3

4

5

6

Time (sec)

Res

idua

l (V

olts

/Met

er)

Figure 38: Plot of the Least Squares residual versus time for the case of constantvariance noise added to the simulated data.

the inverse problem. Again in Figure 37 we note the fan like structure of the residualwith the residual increasing with the absoulte value of the electric field. In the caseof added constant variance noise, however, we again note the lack of a pattern inthe residual as seen in Figure 39. We also observe that there are many points of theresidual on the line Ex = 0. This is beacuse in the simulated data, the electric fieldmagnitude is almost zero most of the time. We compare the simulated data withadded relative random noise and added constant variance noise in Figures 40-43. Wenotice in Figure 43 that the acoustic reflection is drowned by the constant variancenoise with ν = 0.05 (10% RN), which is not the case with the relative random noisedata for the same value of ν. Thus the identification of κs is more difficult withconstant variance noise added to the simulated data.

We next attempt the identification of three parameters, namely εs,0, τ0 and κs. Theother four parameters, ε∞,0, σ, κ∞ and κτ are fixed at values given in (104). Tables 9and 10 present the final estimates and the details of the corresponding Nelder-Meadsimulation. We observe that the final estimates for both τ0 and κs start to increaseaway from their true values as the level of constant variance noise is increased. Theplots of the least squares objective function presented before provide an explanationfor this. In Figures 45-44 we plot the variation of the three parameters over alliterations in the Nelder-Mead simulation. In each of these figures the dashed linerepresents the true value of the corresponding parameter. Again, we observe thatthe estimates for εs,0 are stable with respect to the level of constant variance noise,

64

0 20 40 60 80 100 120 1400

1

2

3

4

5

6

Absolute Value of the Electric Field, |Ex|

Res

idua

l (V

olts

/Met

er)

Figure 39: Plot of the Least Squares residual versus the absolute value of the electricfield for the case of constant variance noise added to the simulated data.

however, the estimates for τ0 and κs increase as the value of ν is increased.

Table 9: Parameter estimation of εs,0 and τ0 and κs forconstant variance noise, corresponding to varying levelsof relative noise with the Nelder-Mead algorithm.

% RN Iter εs,0 τ0(×10−12) κs

0.0 78 80.1069 8.1543 48.20260.1 81 80.1033 8.1627 48.33860.3 75 80.0959 8.1768 48.67250.5 81 80.0890 8.1919 48.99051.0 90 80.0745 8.2242 49.82653.0 79 80.0076 8.3635 53.18105.0 99 79.9315 8.5015 56.792610.0 83 79.7087 8.8299 66.2803

65

1 2 3 4

x 10−9

−100

−50

0.0

50

100

150

Time (sec)

Ele

ctic

Fie

ld D

ata

(Vol

ts/M

eter

)Data with Constant Variance NoiseData with Relative Random Noise

Figure 40: Comparison of the data with added relative random noise to the data withconstant variance noise added. The value of ν in both the cases is ν = 0.05 (10%RN).

1

x 10−9

−100

−50

0.0

50

100

150

Time (sec)

Ele

ctic

Fie

ld D

ata

(Vol

ts/M

eter

)

Data with Constant Variance NoiseData with Relative Random Noise

Figure 41: Comparison of the data with added relative random noise to the data withconstant variance noise added for the input signal at the center of the antenna. Thevalue of ν in both the cases is ν = 0.05 (10% RN).

66

1 1.5 2

x 10−9

−50

−40

−30

−20

−10

0

10

20

30

40

50

Time (sec)

Ele

ctic

Fie

ld D

ata

(Vol

ts/M

eter

)

Data with Constant Variance NoiseData with Relative Random Noise

Figure 42: Comparison of the data with added relative random noise to the data withconstant variance noise added for the first reflection from the air-Debye interface. Thevalue of ν in both the cases is ν = 0.05 (10% RN).

2 2.5 3 3.5 4

x 10−9

−5

−4

−3

−2

−1

0

1

2

3

4

5

Time (sec)

Ele

ctic

Fie

ld D

ata

(Vol

ts/M

eter

)

Data with Constant Variance NoiseData with Relative Random Noise

Figure 43: Comparison of the data with added relative random noise to the data withconstant variance noise added for the reflection from the acoustic wave region. Thevalue of ν in both the cases is ν = 0.05 (10% RN).

67

0 10 20 30 40 50 60 70 80 90 10068

70

72

74

76

78

80

82

Iterations

ε s,0

^

εs,0* = 80.1 (− −)

Figure 44: Estimation of εs,0 by the Nelder-Mead algorithm. We plot the variationof εs,0 with the iteration count.

Table 10: Parameter estimation of εs,0 and τ0 and κs forconstant variance noise, corresponding to varying levelsof relative noise with the Nelder-Mead algorithm.

% RN ||∇JS||L2 FC |JSn+1 − JS

0 | JS

0.0 0.008942 140 8.02867×10−9 4.4391×10−5

0.1 0.00224 144 4.3059×10−9 0.26720.3 0.00499 136 5.9737×10−9 2.40390.5 0.0139 150 3.5686×10−9 6.67711.0 0.001414 161 8.5522×10−9 26.70713.0 0.0277 147 9.0774×10−9 240.35535.0 0.007697 181 6.7874×10−9 667.644110.0 0.001432 154 6.2983×10−9 2670.5056

In the case of the estimation of three parameters we observe that the estimatesof κs are more accurate when τ0 is allowed to vary as opposed to the case of twoparameter estimation when τ0 is fixed at a particular value. Varying τ0, however,does not seem to make much difference in estimating εs,0. Also, comparing the in-tervals in Tables 8 and 11 we find that we obtain tighter confidence intervals for thethree parameter case as opposed to the two parameter case, for the same level of

68

0 10 20 30 40 50 60 70 80 90 10035

40

45

50

55

60

65

70

75

Iterations

κ s^

κs* = 48.06 (− −)

Figure 45: Estimation of κs by the Nelder-Mead algorithm. We plot the variation ofκs with the iteration count.

0 10 20 30 40 50 60 70 80 90 1006.5

7

7.5

8

8.5

9

9.5x 10−12

Iterations

τ

τ

0^

0* = 8.1× 10−12 (− −)

Figure 46: Estimation of τ0 by the Nelder-Mead algorithm. We plot the variation ofτ0 with the iteration count.

69

Table 11: Confidence Intervals for the parameter esti-mation of εs,0,τ0 and κs for constant variance noise withthe Nelder-Mead algorithm.

RN 0.0%εs,0 (8.01069 ± 1.8738 × 10−5) × 10τ0 (8.1543 ± 2.3240 × 10−4) × 10−12

κs (4.82026 ± 2.6832 × 10−4) × 10RN 0.1%εs,0 (8.01033 ± 1.4526 × 10−3) × 10τ0 (8.1627 ± 1.8004 × 10−2) × 10−12

κs (4.83386 ± 2.0798 × 10−2) × 10RN 0.3%εs,0 (8.00959 ± 4.3438 × 10−3) × 10τ0 (8.1768 ± 5.3824 × 10−2) × 10−12

κs (4.86725 ± 6.2220 × 10−2) × 10RN 0.5%εs,0 (8.00890 ± 7.2206 × 10−3) × 10τ0 (8.1919 ± 8.9417 × 10−2) × 10−12

κs (4.89905 ± 1.0345 × 10−1) × 10RN 1.0%εs,0 (8.00745 ± 1.4327 × 10−2) × 10τ0 (8.2242 ± 1.7739 × 10−1) × 10−12

κs (4.98265 ± 2.0555 × 10−1) × 10RN 3.0%εs,0 (8.00076 ± 4.1809 × 10−2) × 10τ0 (8.3635 ± 5.1532 × 10−1) × 10−12

κs (5.31810 ± 6.0160 × 10−1) × 10RN 5.0%εs,0 (7.99315 ± 6.7761 × 10−2) × 10τ0 (8.5015 ± 8.3067 × 10−1) × 10−12

κs (5.67926 ± 9.7751 × 10−1) × 10RN 10.0%εs,0 (7.97087 ± 1.2738 × 10−1) × 10τ0 (8.8299 ± 1.5286) × 10−12

κs (6.62803 ± 1.8366) × 10

70

added noise. The reason for this again is most likely the dependence of the acous-tic reflections on the value of τ0. Thus, the algorithm for the inverse problem canchoose τ0 appropriately so that the value of κs is minimized. As before, estimates forall three parameters become worse as the level of noise added is increased and thecorresponding confidence intervals become larger as seen in Table 11.

In Figure 47 we plot the details of the Nelder-Mead simulation for the identificationof the three parameters εs,0, τ0 and κs. In Figure 48 we plot the least squares objectivefunction for values of τ0 and κs that differ from the true values by up to 10%, inthe case of added constant variance noise with ν = 0.05 (10% RN). The other fiveparameters are fixed at the values given in (104) with εs,0 fixed at the value 73.1. Weobserve that the minimum has moved away from the true values. The fact that theacoustic reflection is overshadowed by the constant variance noise as shown in Figure43 explains this behaviour of the objective function.

6.3 Simulation Results: The Levenberg-Marquardt method

We now repeat our inverse problem with a different least squares optimization tech-nique. The Nelder Mead method presented in the previous section is a gradient freemethod based on the use of simplices. We will now use a gradient based method,namely the Levenberg-Marquardt method to solve our inverse problem. For theoverdetermined least squares objective function

JS(q) =1

2

M∑

k=1

|Ex(tk, 0, zc;q) − Osk|2 =

1

2R(q)T R(q), (110)

Levenberg-Marquardt, which is a trust region method, adds a regularization param-eter νLM > 0 to the approximate Hessian of the Gauss-Newton method to obtain anew estimate qc = qc + s, where

s = −(νLMI + R′(qc)

T R′(qc))−1

R′(qc)T R(qc), (111)

with qc, the current estimate and I the l× l identity matrix, where l is the number ofparameters that we are attempting to estimate. The matrix

(νLMI + R′(qc)

T R′(qc))

is positive definite. The parameter νLM is called the Levenberg-Marquardt parameter.For details of this implementation of the Levenberg-Marquardt method we refer thereader to [Kel99].

As with Nelder-Mead, we first attempt the identification of all the seven parame-ters, then we identify εs,0 and κs and finally we attempt the identification of εs,0, τ0

and κs. Figures 49-50 plot the final estimates for all the seven parameters and thedetails of the Levenberg-Marquardt simulation. The dashed lines here represent thetrue values of the corresponding parameter. Tables 12 and 13 present the results forthe inverse problem of identifying the seven parameters. We observe how close thefinal estimates of the parameters here are with those computed by Nelder-Mead. We

71

0 20 40 60 80 10010−4

10−2

100

102

104

Iterations

L∞ Norm of Gradient

0 20 40 60 80 10010−10

10−5

100

105

Function Differences

0 20 40 60 80 10010−5

100

105

Least Squares Function Value

0 20 40 60 80 10010−5

100

Max Oriented Length

0 20 40 60 80 10010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 20 40 60 80 10010−5

100

105

Average Least Squares value

Figure 47: Nelder-Mead for the three parameter estimation problem for varying levelsof constant variance noise.

72

77.5

88.5

9

x 10−12

JS

40

45

50

5585

95

105

115

0τ

The Least Squares Objective Function

κs

7 7.5 8 8.5 9

x 10−12

85

90

95

100

τ0

JS

40 45 50 5584

86

88

90

92

94

κs

JS

Least Squares Function Cross−Sections

Figure 48: (Top) The Least Squares Objective Function for different values of τ0 andκs. constant variance noise with ν = 0.005 (1% RN) is added to the data. The otherfive parameters were fixed at the values given in (103) with εs,0 fixed at the value 73.1.The minimum is located at the point (8.91× 10−12, 43.254), with a minimum value of85.796. (Bottom) Plot of the Least Squares Objective function versus κs (left) andversus τ0 (right).

73

0 5 10 1510−4

10−2

100

102

104

Iterations

Norm of Gradient

0 5 10 1510−6

10−4

10−2

100

102

Iterations

Least Squares Function Value

Figure 49: Levenberg-Marquardt for the seven parameter estimation problem

0 5 10 1576

77

78

79

80

81

Iterations

ε s,0

^

εs,0* = 80.1

0 5 10 1544

45

46

47

48

49

Iterations

κ s^

κs* = 48.06

0 5 10 157.6

7.7

7.8

7.9

8

8.1x 10−12

Iterations

τ

τ

0^

0

* = 8.1e−12

Figure 50: Estimation of εs,0, τ0 and κs by the Levenberg-Marquardt algorithm. Weplot the variation of the three parameters with the iteration count.

74

0 5 10 155.0

5.2

5.4

5.6

ε ∞,0

^

ε∞,0* = 5.5 (− −)

0 5 10 150.9

0.95

1.0

1.05

1.1

σ

σ

^

* = 1.0e−5 (− −)

0 5 10 154.1

4.2

4.3

4.4

4.5

Iterations

κ ∞^

κ∞* = 4.4 (ε∞) (− −)

0 5 10 153.8

3.9

4

4.1x 10−13

Iterations

κ τ^

κτ* = 4.05e−13 (− −)

Figure 51: Levenberg-Marquardt for the seven parameter estimation problem. Weplot the variation of the parameters with the iteration count. The dashed lines in-dicate the true values of the parameters, while the solid lines are the estimates overthe iterations

75

Table 12: Final estimates for all seven parameters for water using Levenberg-Marquardt. The initial parameter set has values that are 5% smaller than the truevalues.

Parameter True Values Final Estimatesεs,0 80.1 80.0917ε∞,0 5.5 5.2039τ0 8.1e − 12 8.0666 × 10−12

σ 1.0 × 10−5 9.5 × 10−6

κs 48.06 48.0028κ∞ 4.4 4.1786κτ 4.05 × 10−13 3.8462 × 10−13

Table 13: Status of the Levenberg-Marquardt Simulation at the final it-eration.

Least squares value 1.0584 × 10−5

L2 norm of the gradient 5.7248 × 10−4

Number of forward solves 31Number of gradient evaluations 16Number of hessian evaluations 30

can again make similar conclusions as we did in the case of the Nelder-Mead results.Only εs,0, τ0 and κs seem to be identifiable with some accuracy, whereas the problemappears to be insensitive to changes in the other four parameters. We next attemptthe identification of εs,0 and κs with simulated data in which constant variance noiseis added. The values of the remaining five parameters are fixed at those in (104).Tables 14 and 15 present the final estimates and the details of the corresponding LMsimulation. In Figures 52 and 53 we plot the details of the LM run as well as thevariation of εs,0 and κs over all iterations. The confidence intervals for these estimatesare presented in Table 16. Again, we can make similar remarks and observations asin the case of the results of the Nelder-Mead method.

Finally we attempt to identify the three parameters εs,0, τ0 and κs. Figures 55 and54 plot the details of the LM simulation and the variation of the parameters over alliterations. Tables 17 and 18 present the final estimates and details of the algorithm.Again, as was observed in the results of the Nelder-Mead runs, the estimates for κs

are better when the value of τ0 is kept variable as opposed to fixing the value of τ0.Thus, the inverse problem captures κs more accurately if the value of τ0 is allowed tochange. In Table 19 we present the confidence intervals for the three parameter

76

Table 14: Parameter estimation of εs,0 and κs for constant variance noise, correspond-ing to varying levels of relative noise with the Levenberg-Marquardt algorithm.

% RN Iter εs,0 κs JS ||∇JS||L2

0.0 18 80.2166 55.2951 0.4369 0.00065260.5 18 80.191 55.8726 7.0804 0.00711.0 18 80.1654 56.4524 27.0778 0.000763.0 18 80.0636 58.7948 240.6053 0.0010685.0 18 79.9621 61.1765 667.7927 0.00138510.0 19 79.7084 67.3217 2670.5115 0.001417

Table 15: Parameter estimation of εs,0 and κs for constant variance noise, correspond-ing to varying levels of relative noise with the Levenberg-Marquardt algorithm.

% RN Iter JS ||∇JS||L2 F G H0.0 18 0.4369 0.0006526 37 19 360.5 18 7.0804 0.0071 37 19 361.0 18 27.0778 0.00076 37 19 363.0 18 240.6053 0.001068 37 19 365.0 18 667.7927 0.001385 37 19 3610.0 19 2670.5115 0.001417 39 20 38

0 5 10 15 2072

74

76

78

80

82

Iterations

ε s,0

^

εs,0* = 80.1

0 5 10 15 2040

45

50

55

60

65

70

Iterations

κ s^

κs* = 48.06

Figure 52: Levenberg-Marquardt for the two parameter estimation problem. Varia-tion of εs,0 (left) and κs (right) over the iteration count.

77

Table 16: Confidence Intervals for the parameter estimation of εs,0 and κs for constantvariance noise with the Levenberg-Marquardt algorithm.

RN = 0.0%εs,0 (8.02166 ± 1.8803 × 10−3) × 10κs (5.52951 ± 1.6448 × 10−2) × 10

RN = 0.5%εs,0 (8.01910 ± 7.5113 × 10−3) × 10κs (5.58726 ± 6.5668 × 10−2) × 10

RN = 1.0%εs,0 (8.01654 ± 1.4577 × 10−2) × 10κs (5.64524 ± 1.2738 × 10−1) × 10

RN = 3.0%εs,0 (8.00636 ± 4.2160 × 10−2) × 10κs (5.87948 ± 3.6783 × 10−1) × 10

RN = 5.0%εs,0 (7.99621 ± 6.8206 × 10−2) × 10κs (6.11765 ± 5.9447 × 10−1) × 10

RN = 10.0%εs,0 (7.97084 ± 1.2723 × 10−1) × 10κs (6.73217 ± 1.1076) × 10

0 5 10 15 2010−4

10−2

100

102

104

Iterations

Norm of Gradient

0 5 10 15 2010−1

100

101

102

103

104

Iterations

Least Squares Function Value

Figure 53: Estimation of εs,0 and κs by the Levenberg-Marquardt algorithm for vary-ing levels of constant variance noise. Details of the LM simulation.

78

0 5 10 15 2072

74

76

78

80

82

Iterations

ε s,0

^

εs,0* = 80.1 (− −)

0 5 10 15 207.2

7.6

8.0

8.4

8.8

9.2

Iterations

τ 0^

τ 0* = 8.1× 10−12 (− −)

0 5 10 15 2040

50

60

70

Iterations

κ s^

κs* = 48.06 (− −)

Figure 54: Estimation of εs,0, τ0 and κs by the Levenberg-Marquardt algorithm withvarying levels of added constant variance noise. We plot the variation of the threeparameters with the iteration count.

0 5 10 15 2010−4

10−2

100

102

104

Iterations

Norm of Gradient

0 5 10 15 2010−6

10−4

10−2

100

102

104

Iterations

Least Squares Function Value

Figure 55: Levenberg-Marquardt for the three parameter estimation problem withvarying levels of added constant variance noise. Details of the LM simulation.

79

Table 17: Parameter estimation of εs,0 and τ0 and κs for constant variance noise,corresponding to varying levels of relative noise with the Levenberg-Marquardt algo-rithm.

% RN Iter εs,0 τ0(×10−12) κs JS ||∇JS||L2

0.0 17 80.1083 8.1562 48.2113 4.4383×10−5 0.00096371.0 17 80.0765 8.2247 49.8060 26.7071 0.0013713.0 18 80.0079 8.3618 53.1392 240.3553 0.0013455.0 19 79.9320 8.4983 56.6654 667.6441 0.000804410.0 20 79.7085 8.8330 66.3476 2670.5056 0.001268

Table 18: Parameter estimation of εs,0 and τ0 andκs for constant variance noise, corresponding tovarying levels of relative noise with the Levenberg-Marquardt algorithm.

% RN Iter JS ||∇JS||L2 F G H0.0 17 4.4383×10−5 0.0009637 35 18 341.0 17 26.7071 0.001371 35 18 343.0 19 240.3553 0.001345 37 19 365.0 19 667.6441 0.0008044 39 20 3810.0 20 2670.5056 0.001268 41 21 40

80

Table 19: Confidence Intervals for the parameter esti-mation of εs,0,τ0 and κs for constant variance noise withthe Levenberg-Marquardt algorithm.

RN 0.0%εs,0 (8.01083 ± 1.8741 × 10−5) × 10τ0 (8.1562 ± 2.3234 × 10−4) × 10−12

κs (4.82113 ± 2.6830 × 10−4) × 10RN 1.0%εs,0 (8.00765 ± 1.4334 × 10−2) × 10τ0 (8.2247 ± 1.7742 × 10−1) × 10−12

κs (4.98060 ± 2.0560 × 10−1) × 10RN 3.0%εs,0 (8.00079 ± 4.1822 × 10−2) × 10τ0 (8.3618 ± 5.1552 × 10−1) × 10−12

κs (5.31392 ± 6.0178 × 10−1) × 10RN 5.0%εs,0 (7.99320 ± 6.7841 × 10−2) × 10τ0 (8.4983 ± 8.3159 × 10−1) × 10−12

κs (5.66654 ± 9.7843 × 10−1) × 10RN 10.0%εs,0 (7.97085 ± 1.2736 × 10−1) × 10τ0 (8.8330 ± 1.5277) × 10−12

κs (6.63476 ± 1.8360) × 10

estimation problem.

7 A second test case

In this section we consider the parameter estimation for a Debye medium with a largerrelaxation time than the value that was considered in the first test problem. We wouldlike to study the effect of the relaxation time τ and the interrogation frequency ωc onthe ability of the electromagnetic pulse to penetrate the dielectric material. As shownin [BBL00] a very small relaxation time, such as that for water τ ∗

0 = 8.1× 10−12, willmake the material appear comparatively “hard” in the sense that it will allow muchless of the signal to penetrate the air-Debye interface at z = z1 for a sufficiently highinterrogation frequency like ωc = π × 1010 rad/sec. This small transmitted wave canonly generate very little, if any, reflection at the second interface as it enters theregion containing the pressure wave. This demonstrates the importance of the choiceof the carrier frequency ωc for effective interrogation. In [BBL00] the authors have

81

demonstrated the strong influence of the dielectric relaxation time on an appropriatechoice of the interrogating frequency for successfully penetrating the material. In thissecond test case we consider a parameter estimation problem for a Debye medium witha relaxation time τ ∗

0 = 3.16×10−8 (as used in [BBL00, ABR02]), and an interrogationfrequency ωc = π × 1010 rad/sec.

As observed in Section (4.1), the product ωcτ0 strongly determines the parametersin the problem that can be identified accurately. We will observe that by changingthe values of τ0 and ωc in this second test problem, the parameters that dominate inthe term ε∗r in (74) are no longer the same as those of the first test problem, i.e., ε∗s,0and κ∗

s. As will be demonstrated in Section (7.2), the parameters whose magnitudedominates the quantity ε∗r are ε∗∞,0 and the pressure coefficient κ∗

∞.The data that we will attempt to fit is generated with a Debye model which is

defined by the parameter values

ε∗s,0 = 78.2

ε∗∞,0 = 5.5

τ ∗0 = 3.16 × 10−8

σ∗ = 1 × 10−5

κ∗τ = 0.05τ ∗

0 = 1.58 × 10−9

κ∗s = 0.6ε∗s,0 = 46.92

κ∗∞ = 0.3ε∗∞,0 = 1.65.

(112)

In this model, the value of τ ∗0 is much higher than in the case of the first test problem.

As will be seen in the sequel, this value of τ ∗0 leads to a parameter estimation problem

with very different properties.

7.1 The forward problem: Simulation results

We perform similar numerical simulations, as were done for the first test problem,to collect simulated data at the center of the antenna and to demonstrate the effectof the acoustic pressure on the electric field. We would like to know how and towhat extent the acoustic pressure can change the reflected electric wave from theinterface at z = z2. In Figure 56 we plot the electromagnetic source that is used inour simulations. The form for the source is

Js(t, x, z) = I(x1,x2)δ(z) sin(ωc(t − 3t0)) exp

(−[t − 3t0

t0

]2)

x,

t0 =1

π × 109; ωc = π × 1010 rad/sec; fc = 5.0 × 109 Hz.

(113)

where I(x1,x2) is the Indicator function on (x1, x2). The Fourier spectrum of this pulsehas even symmetry about 5.0 GHz. In Figure 57 we plot the power spectral densityof the source defined in (113). We see that the power spectral density is symmetricabout 5.0 GHz.

The computational domain is defined as follows. We take X0 = (0, 0.12), Za =(0, 0.33) and ZD = (0.33, 0.5). The number of nodes along the z-axis is taken to be 450

82

and the number of nodes along the x-axis is taken to be 108. The spatial step size inboth the x and z directions is ∆x = ∆z = h = 0.06/54. From the CFL condition (55)with the Courant number ηCN = 1/2 we obtain the time increment to be ∆t ≈ 1.852pico seconds. The central frequency of the input source as described in (54) is 5.0 GHzand thus the wavelength is 0.06 meters. The antenna is half a wavelength long and isplaced at (x1, x2)×zc, with zc = 0, x1 = 0.03 and x2 = 0.09. We use PML layers thatare half a wavelength thick on all fours sides of the computational domain as shownin Figure 3. The reflections of the electromagnetic pulse at the air-Debye interfaceand from the acoustic pressure wave are recorded at the center of the antenna (xc, zc),with xc = 0.06, at every time step. This data will be used as observations for theparameter identification problem to be presented later. The component of the electricfield that is of interest here is the Ex component. Thus our data is the set

E(q∗) = Ex(t = n∆t, xc, zc;q∗)M

n=1

q∗ = (ε∗s,0, ε∗∞,0, τ

∗0 , σ∗, κ∗

s, κ∗∞, κ∗

τ )T .

(114)

The windowed acoustic pressure wave as defined in (49) has the parameter valuesωp = 4.0π × 104 rad/sec, cp = 667.67 m/s, and thus, λp = 0.033. The location of thepressure region is in the interval (z2, z2 + λp) = (0.44, 0.47).

In Figures 58 -61 we plot the Ex field magnitude in the plane containing the centerof the antenna versus depth along the z axis. Figure 58 depicts the electromagneticwave penetrating the Debye medium. In 59 we see the reflection of the electromagneticwave from the Debye medium moving towards the antenna and the Brillouin precursorpropagating in the Debye medium. In Figure 60 we observe the reflection from theregion containing the acoustic pressure moving towards the air-Debye interface andthe transmitted part of the electromagnetic source travelling into the absorbing layer.The reflection from the acoustic pressure crosses the air-Debye interface in Figure 61.Here we see another reflected wave from the air-Debye interface that moves towardsthe right. In Figures 62 and 63 we plot the Ex field magnitude recorded at the centerof the antenna versus time, which shows the electromagnetic source, the reflectionoff the air-Debye interface and the reflection from the region containing the acousticpressure wave. As seen here the magnitude of the reflection from the acoustic pressureis many orders of magnitude smaller that the initial electromagnetic source as wellthe reflection from the air-Debye interface.

7.2 Sensitivity analysis

Here we examine the system dynamics as the parameters vary. We are interested inthe changes produced by the coefficents of pressure in the polarization, namely theparameters κs, κ∞ and κτ . In Figure 65 we plot the difference in the Ex componentof the electric field between the simulation with parameter values given in (112)and simulations for which the value of κ∞ is 0, 0.4ε∞,0 and 0.5ε∞,0. As expectedthe difference is seen in the acoustic reflection that is observed at the center of the

83

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10−9

−200

−150

−100

−50

0

50

100

150

200

Time (sec)

Sou

rce

(Vol

ts/m

eter

)

Figure 56: Forward Simulation for a Debye medium with parameters given in (112).The input electromagnetic source as defined in (113).

antenna. In Figure 66 we plot the difference in the Ex component of the electricfield between the simulation with parameter values given in (112) and simulations forwhich the value of κs is 0, and 0.3εs,0. In Figure 67 we plot the difference in the Ex

component of the electric field between the simulation with parameter values given in(112) and simulations for which the value of κτ is 0 and 0.1εs,0. Again, in both casesthe difference is seen in the acoustic reflection that is observed at the center of theantenna. We observe that the difference in the Ex field in the case of Figures 66 and67 is one or two orders of magnitude smaller than that in Figure 65. This indicatesthat the coefficent that is most significant for this test problem and which will beestimated more accurately than the others is κ∞, the pressure coefficient in ε∞. Inthe first test problem κs, the pressure coefficient in εs was the most significant. Sowe see a very different behaviour of the acoustic reflection in this case.

This observation, as was done for the first test problem, can also be deduced froman analysis of (40). In the simulations for this test problem the outgoing and reflectedradiation will be dominated by frequencies near the center frequency 5.0 GHz. Thus,ε∗r will be dominated by frequencies near 5.0 GHz. In this problem ωτ ∗

0 ≈ O(102),

84

3 5 9 11 13

x 109

0

1

2

3

4

5x 108

Frequency (Hz)

|Y|2

The power spectral density of the source

Figure 57: Fourier transform of the source defined in (113). The transform is centeredaround the central frequency of ω = π × 1010 rad/sec.

and with ε0 = 8 × 10−12 we have (compare with the estimates of (75))

(ε∗s,0

1 + jωτ ∗0

)≈ O(10−2ε∗s,0) ≈ O(10−1),

(jωτ ∗

0

1 + jωτ ∗0

)ε∗∞,0 ≈ O(ε∗∞,0) ≈ O(10),

(σ∗

jωε0

)≈ O(103σ∗) ≈ O(10−3).

(115)

Thus we can see that ε∗r will be most sensitive to the infinite frequency permittivityε∞,0 and the effects of εs,0 and τ0 and σ will not be as pronounced. Also, this impliesthat ε∗r will be more sensitive to the pressure coefficient κ∞ than to the coefficientsκs and κτ , which is the same observation that was made from Figures 65, 66 and 67

We next study the effect that the acoustic speed and frequency have on the am-plitude of the reflection from the acoustic pressure region as was done in the firsttest problem. In Figure 68 we plot the acoustic reflection observed at the center ofthe antenna for different values of the acoustic frequency and in Figure 69 we plotthe acoustic reflection for different values of the acoustic speed. As noted before, bychanging the speed or the frequency, the wavelength λp changes and thus the sizeof the interval (z2, z2 + λp) in which the pressure wave is generated. The amplitudeof the acoustic reflection appears to increase as the acoustic frequency is increasedand decreases as the acoustic speed increases, the opposite behaviour to what was

85

0 0.1 0.2 0.33 0.44 0.47 0.5−50

−40

−30

−20

−10

0

10

20

30

40

50

z (meters)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

t = 1.85 × 10−9 sec

Air Debye Medium

Pressure wave region

Figure 58: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude along the plane containing the center of the antenna versusdepth. We see here the electromagnetic source penetrating the Debye medium.

0 0.1 0.2 0.33 0.44 0.47 0.5−50

−40

−30

−20

−10

0

10

20

30

40

50

z (meters)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

t = 2.78 × 10−9 sec

Air Debye Medium

Pressure wave region

Figure 59: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude along the plane containing the center of the antenna versusdepth. The plot shows the reflection of the electromagnetic source off the air-Debyeinterface travelling towards the antenna. We also see the brillouin precursor propa-gating inside the Debye medium.

86

0 0.1 0.2 0.33 0.44 0.47 0.5−50

−40

−30

−20

−10

0

10

20

30

40

50

z (meters)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

t = 3.7 × 10−9 sec

Air Debye Medium

Pressure wave region

Figure 60: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude along the plane containing the center of the antenna versusdepth. Here we see the reflection from the acoustic pressure wave moving towardsthe left.

0 0.1 0.2 0.33 0.44 0.47 0.5−50

−40

−30

−20

−10

0

10

20

30

40

50

z (meters)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

t = 4.63 × 10−9 sec

Air Debye Medium

Pressure wave region

Figure 61: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude along the plane containing the center of the antenna. Thereflection from the acoustic pressure wave crosses the air-Debye interface. A secondaryreflection off this interface travels towards the right.

87

0 1 2 3 4 5 6

x 10−9

−150

−100

−50

0

50

100

150

Time t (secs)

Ex fi

eld

mag

nitu

de (V

olts

/met

er) Debye Interface

reflection Acoustic Interfacereflection

Figure 62: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude versus time. We see three sets of observations. The electro-magnetic source, the reflection off the air-Debye interface and the relection from theacoustic pressure wave.

2 3 4 5 6

x 10−9

−10

−5

0

5

10

Time t (secs)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

Debye Interfacereflection

Acoustic Interfacereflection

Figure 63: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude versus time. Here we magnify the two sets of reflections, onefrom the air-Debye interface and from the acoustic pressure wave.

88

3 5 7 9 11 13 15

x 109

0

1

2

3

4

5

x 108

Frequency (Hz)

|Y|2

The power spectral density of the observed Ex field

Figure 64: Fourier transform of the electric field data in Figure 62. The transform iscentered around the central frequency of ω = 10π × 109.

4 5 6

x 10−9

−3

−2

−1

0

1

2

3

Time t (secs)

Ex fi

eld

mag

nitu

de (V

olts

/met

er)

κ∞ = 0.0

κ∞ = 0.3 ε

∞,0κ

∞ = 0.4 ε

∞,0κ

∞ = 0.4 ε

∞,0

Figure 65: Forward Simulation for a Debye medium with parameters given in (112).The Ex field magnitude measured at the center of the antenna for different values ofκ∞.

89

0 1 2 3 4 5 6

x 10−9

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time t (secs)

Diff

eren

ce in

Ex F

ield

Mag

nitu

de

κs = 0.6ε

s,0 , κ

s = 0

κs = 0.6ε

s,0 , κ

s = 0.3ε

s,0

Figure 66: Difference in the Ex field magnitude measured at the center of the antennabetween the simulation for parameters given in (112) and simulations with differentvalues of κs.

observed in the first test problem. We note here again that as our windowed pressurewave contains only one wavelength of the sinusoid it is difficult to predict how thereflections will behave as the speed or the frequency is changed.

7.3 Simulation results for the inverse problem: The Leven-

berg Marquardt method

In this section we present results for the inverse problem of parameter estimation forthe second test problem. We do this first with the Levenberg-Marquardt algorithmand later with the Nelder-Mead method. We first attempt to estimate the threeparameters ε∞,0, τ0 and the pressure coefficient κ∞, of ε∞. The values of the remainingparameters are fixed at

ε∗s,0 = 71.09 (relative static permittivity),

ε∗∞,0 = 5.5 (relative high frequency permittivity),

τ ∗0 = 3.48 × 10−8 seconds,

σ∗ = 1.5 × 10−5 (mhos/meter),

(116)

90

0 1 2 3 4 5 6

x 10−9

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10−3

Time t (secs)

Diff

eren

ce in

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

κτ = 0.05τ

0 , κ

τ = 0

κτ = 0.05τ

0 , κ

τ = 0.1τ

0

Figure 67: Difference in the Ex field magnitude measured at the center of the antennabetween the simulation for parameters given in (112) and simulations with differentvalues of κτ .

κ∗s = 42.654 (pressure coefficient in ε∗s),

κ∗∞ = 1.65 (pressure coefficient in ε∗∞),

κ∗τ = 1.74 × 10−9 (pressure coefficient in τ ∗).

(117)

The results are tabulated in Tables 20 and 21 and presented in Figures 70-73. Wehave added constant variance noise with different values of the scale parameter ν tothe data. We note from Table 20 that the estimation of ε∞,0 is relatively stable overall values of ν that we have selected; whereas the estimation of κ∞ becomes inaccurateas the value of ν is increased. Unlike the previous test problem, the estimation of τ0 inthis case is not good. Table 21 presents the number of function, gradient and hessianevaluations that are required for the algorithm to converge, where the convergencecriteria used here is ||∇JS

k ||/||∇JS0 || < 10−6.

In Table 22 we present the confidence intervals for the three parameter estimationproblem with results tabulated in Table 20. Again, we note that the confidenceintervals become larger as the level of noise increases. From Table 22 we observe thatthe confidence level in the estimates ε∞ and κ∞ is higher than the confidence level inτ0, as is also observed from the final estimates for τ0.

We now attempt to estimate the two parameters ε∞,0 and its pressure coefficientκ∞. We fix the value of the relaxation time τ0 to be

τ0 = 3.48 × 10−8. (118)

91

0.4 0.42 0.44 0.46 0.48 0.5−1

−0.5

0

0.5

1

z (meters)

Win

dow

ed A

cous

tic P

ress

ure

Wav

e

fp = 4.0π × 104 rad/sec

fp = 6.0π × 104 rad/sec

fp = 8.0π × 104 rad/sec

fp = 12.0π × 104 rad/sec

4 4.5 5 5.5 6

x 10−9

−4

−3

−2

−1

0

1

2

3

4

Time t (secs)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

fp = 4.0π × 104 rad/sec

fp = 6.0π × 104 rad/sec

fp = 8.0π × 104 rad/sec

fp = 12.0π × 104 rad/sec

Figure 68: Plot of the magnitude of the acoustic reflection against time for differentvalues of the acoustic frequency. The other parameters are fixed at their true valuesgiven in (112).

92

0.4 0.42 0.44 0.46 0.48 0.5−1

−0.5

0

0.5

1

z (meters)

Win

dow

ed A

cous

tic P

ress

ure

Wav

e

cp = 400 m/s

cp = 667 m/s

cp = 800 m/s

cp = 1000 m/s

4 4.5 5 5.5 6

x 10−9

−2.5

−1.5

−0.5

0

0.5

1.5

2.5

z (meters)

Ex F

ield

Mag

nitu

de (V

olts

/met

er)

c = 400 m/sc = 667 m/sc = 800 m/sc = 1000 m/s

pppp

Figure 69: Plot of the magnitude of the Ex component of the electric field againsttime for different values of the acoustic speed. The other parameters are fixed at theirtrue values given in (112).

93

Table 20: Parameter estimation of ε∞,0, τ0 and κ∞ for constant variance noise, corre-sponding to varying levels of relative noise with the Levenberg-Marquardt algorithm.

% RN Iter ε∞,0 τ0 κ∞ JS ||∇JS||L2

0.0 32 5.4999 2.8522 × 10−8 1.6496 2.1605 × 10−6 0.00027460.1 32 5.4998 2.8761 × 10−8 1.6463 0.2749 0.00027230.3 32 5.4996 2.9253 × 10−8 1.6395 2.4743 0.00034170.5 32 5.4993 2.9764 × 10−8 1.6328 6.8730 0.00047031.0 34 5.4985 3.1135 × 10−8 1.6160 27.4921 0.00022613.0 39 5.4956 3.8493 × 10−8 1.5467 247.4282 0.00024415.0 45 5.4926 5.1264 × 10−8 1.4753 687.2987 0.000307310.0 57 5.4852 4.1950 × 10−7 1.2957 2749.1770 0.00007404

Table 21: Parameter estimation of ε∞,0, τ0 and κ∞ for constant variance noise, corre-sponding to varying levels of relative noise with the Levenberg-Marquardt algorithm.

% RN Iter JS ||∇JS||L2 F G H0.0 32 2.1605 × 10−6 0.0002746 71 33 700.1 32 0.2749 0.0002723 71 33 700.3 32 2.4743 0.0003417 71 33 700.5 32 6.8730 0.0004703 71 33 701.0 34 27.4921 0.0002261 76 35 753.0 39 247.4282 0.0002441 86 40 855.0 32 687.2987 0.0003073 98 46 9710.0 57 2749.1770 0.00007404 136 58 135

94

0 10 20 30 40 50 6010−6

10−4

10−2

100

102

104

Iterations

Norm of Gradient

0 10 20 30 40 50 6010−6

10−4

10−2

100

102

104

Iterations

Least Squares Function Value

Figure 70: Parameter estimation of ε∞,0, τ0 and κ∞ using the Levenberg-Marquardtalgorithm with added constant variance noise for various values of ν.

0 10 20 30 40 50 605.1

5.3

5.5

5.7

5.9

6.1

Iterations

ε ∞,0

^

ε∞,0* = 5.5

Figure 71: Levenberg-Marquardt for the three parameter estimation problem withadded constant variance noise. Variation of the estimate ε∞,0 with the iterationcount.

95

0 10 20 30 40 500

1

2

3

4

5x 10

−7

Iterations

τ

τ

^ 0

0* = 3.16× 10−8

0 10 20 30 40 502.5

3.5

4.5

5.5x 10

−8

Iterations

τ 0

0τ* = 3.16 × 10−8

Figure 72: Levenberg-Marquardt for the three parameter estimation problem withadded constant variance noise. We plot the variation of the estimate τ0 with iterationcount (top) and a magnified plot of the same (bottom).

96

Table 22: Confidence Intervals for the parameter estima-tion of ε∞,0, τ0 and κ∞ with the Levenberg-Marquardtalgorithm for varying levels of constant variance noisecorresponding to 0%-10% relative noise.

RN = 0.0%ε∞,0 5.4999 ± 9.8421 × 10−7

τ0 (2.8522 ± 1.9957 × 10−4) × 10−8

κ∞ 1.6496 ± 2.8669 × 10−5

RN = 0.1%ε∞,0 5.4998 ± 3.5104 × 10−4

τ0 (2.8761 ± 7.1807 × 10−2) × 10−8

κ∞ 1.6463 ± 1.0238 × 10−2

RN = 0.3%ε∞,0 5.4996 ± 1.0531 × 10−3

τ0 (2.9253 ± 2.1927 × 10−1) × 10−8

κ∞ 1.6395 ± 3.0784 × 10−2

RN = 0.5%ε∞,0 5.4993 ± 1.7548 × 10−3

τ0 (2.9764 ± 3.7209 × 10−1) × 10−8

κ∞ 1.6328 ± 5.1417 × 10−2

RN = 1.0%ε∞,0 5.4985 ± 3.5076 × 10−3

τ0 (3.1135 ± 7.7975 × 10−1) × 10−8

κ∞ 1.6160 ± 1.0340 × 10−1

RN = 3.0%ε∞,0 5.4956 ± 1.0504 × 10−2

τ0 (3.8493 ± 2.9130) × 10−8

κ∞ 1.5467 ± 3.1752 × 10−1

RN = 5.0%ε∞,0 5.4926 ± 1.7460 × 10−2

τ0 (5.1264 ± 6.5133) × 10−8

κ∞ 1.4753 ± 5.4261 × 10−1

RN = 10.0%ε∞,0 5.4852 ± 3.4394 × 10−2

τ0 (4.1950 ± 10.3835) × 10−8

κ∞ 1.2957 ± 11.5944 × 10−1

97

0 10 20 30 40 50−0.5

0.5

1.65

Iterations

κ ∞^

κ∞* = 1.65

Figure 73: Levenberg-Marquardt for the three parameter estimation problem withadded constant variance noise. We plot the variation of the estimate κ∞ with theiteration count.

In Figures 74and 75 we plot the two parameter estimation inverse problem with noadded noise to the simulated data. In Figures 76 and 77 we plot the two parameterestimation inverse problem with added constant variance noise to the simulated data.In Tables 23 and 24 we present the final estimates and details of the Levenberg-Marquardt simulations with varying levels of constant variance noise added to thedata that is to be fit. In Table 25 we calculate the confidence intervals for thetwo parameter estimation problem of estimating ε∞,0 and κ∞ with added constantvariance noise to the simulated data. In Figure 78 we plot the least squaresobjective function for different values of ε∞,0 and κ∞. There is no noise added to thedata for these plots. The other five parameters are fixed at their true values as givenin (112). We see possibilities of local minima in these plots. This test problem hasmany local minima whose evidence will be seen clearly in later plots. In Figure 79 weplot the least squares objective function for various values of the two parameters ε∞,0

and κ∞. Again we have not added any noise to the data for these plots. The otherfive parameters were fixed at values given in (116) and (117) except for τ0 which wasfixed at its true value as given in (112).

98

0 10 20 3010−5

100

105

Iterations

Norm of Gradient

0 10 20 3010−2

100

102

104

Iterations

Least Squares Function Value

Figure 74: Parameter estimation of ε∞,0 and κ∞ with no added noise using theLevenberg-Marquardt algorithm. The plot shows the details of the Levenberg-Marquardt algorithm.

0 10 20 302.5

3.5

4.5

5.5

6.5

Iterations

ε ∞,0

^

ε∞,0* = 5.5

0 10 20 30−0.5

0.5

1.5

2.5

Iterations

κ ∞^

κ∞* = 1.65

Figure 75: Levenberg-Marquardt for the two parameter estimation problem with noadded noise. We plot the variation of ε∞,0 (left) and κ∞ (right) versus the iterationcount.

99

0 10 20 30 40 504.8

5

5.2

5.4

5.6

5.8

Iterations

ε ∞,0

^

ε∞,0* = 5.5

0 10 20 30 40 50−1.5

−1

−0.5

0

0.5

1

1.5

2

Iterations κ ∞^

κ∞* = 1.65

Figure 76: Parameter estimation of ε∞,0 and κ∞ with the Levenberg-Marquardt al-gorithm for varying levels of constant variance noise (ν), corresponding to 0%-10%relative noise. We plot the variation of ε∞,0 (left) and κ∞ (right) versus the iterationcount.

0 10 20 30 40 5010−4

10−2

100

102

104

Iterations

Norm of Gradient

0 10 20 30 40 5010−1

100

101

102

103

104

Iterations

Least Squares Function Value

Figure 77: Details of the Levenberg-Marquardt simulations for the two parameterestimation problem with varying levels (ν) of constant variance noise added to thesimulated data as tabulated in Table 23.

100

Table 23: Parameter estimation of ε∞,0 and κ∞ withthe Levenberg-Marquardt algorithm for varying levels ofconstant variance noise, corresponding to 0%-10% rela-tive noise.

% RN Iter ε∞,0 κ∞ JS ||∇JS||L2

0.1 30 5.4988 1.5708 0.2924 0.00039250.3 30 5.4986 1.5714 2.4885 0.00038610.5 30 5.4984 1.5721 6.8843 0.00037681.0 29 5.4979 1.5738 27.4975 0.00074383.0 44 5.4961 1.5806 247.4317 0.00037015.0 23 5.4943 1.5873 687.3371 0.000294810.0 24 4.9006 -1.0116 2800.2627 0.0006955

Table 24: Parameter estimation of ε∞,0 and κ∞ withthe Levenberg-Marquardt algorithm for varying levelsof constant variance noise, corresponding to 0%-10% ofrelative noise.

% RN Iter JS ||∇JS||L2 F G H0.1 30 0.2924 0.0003925 65 31 640.3 30 2.4885 0.0003861 65 31 640.5 30 6.8843 0.0003768 65 31 641.0 29 27.4975 0.0007438 63 30 623.0 44 247.4317 0.0003701 91 45 905.0 23 687.3371 0.0002948 49 24 4810.0 24 2800.2627 0.0006955 52 25 51

101

Table 25: Confidence Intervals for the pa-rameter estimation of ε∞,0 and κ∞ with theLevenberg-Marquardt algorithm for varyinglevels (ν) of constant variance noise corre-sponding to 0%-10% relative noise.

RN = 0.1%ε∞,0 5.4988 ± 3.3500 × 10−4

κ∞ 1.5708 ± 2.0666 × 10−3

RN = 0.3%ε∞,0 5.4986 ± 9.7663 × 10−4

κ∞ 1.5714 ± 6.0237 × 10−3

RN = 0.5%ε∞,0 5.4984 ± 1.6231 × 10−3

κ∞ 1.5721 ± 1.0009 × 10−2

RN = 1.0%ε∞,0 5.4979 ± 3.2375 × 10−3

κ∞ 1.5738 ± 1.9955 × 10−2

RN = 3.0%ε∞,0 5.4961 ± 9.6379 × 10−3

κ∞ 1.5806 ± 5.9286 × 10−2

RN = 5.0%ε∞,0 5.4943 ± 1.5943 × 10−2

κ∞ 1.587 ± 9.7880 × 10−2

102

JS

4.5

5

5.5

6

6.5

1.4

1.5

1.6

1.7

1.8

1.90

100

200

300

400

500

600

ε∞,0

The Least Squares Objective Function

κ∞

4.5 5.5 6.50

250

500

ε∞,0

JS

1.45 1.65 1.850

1

2

3

κ∞

JS

Least Squares Objective Function Cross−Sections

Figure 78: (Top) The Least Squares Objective Function for different values of ε∞,0

and κ∞ with no noise added to the data. The other five parameters were fixed attheir true values as given in (112). The minimum is located at the point (5.5, 1.65),with a minimum value of 0. (Bottom) Cross Sections of the Least Squares Objectivefunction versus κ∞ (right) and versus ε∞,0 (left).

103

In Figure 80 we plot the least squares objective function for a range of values ofε∞,0 and κ∞, again with no added noise. In this case we have not added any noise tothe data. The other five parameters were fixed at values given in (116), and (117),and τ0 fixed at the value given in (118) that have 10-50% relative error as comparedwith their true values in (112).

7.4 Simulation results for the inverse problem: The Nelder-

Mead method

We first attempt to estimate the two parameters ε∞,0 and its pressure coefficient κ∞.We add constant variance noise to our simulated data. The other five parameters arefixed at their values given in (116), and (117) with τ0 fixed at the value given in (118).Figures 81 and 82 plot the details of the Nelder-Mead simulations and the variation ofthe two parameters over all iterations. Table 26 displays the final estimates and thedetails of the Nelder-Mead algorithm for the two parameter estimation problem withadded constant variance noise. We note that the Nelder-Mead algorithm does notconverge to the optimal values of either of the two parameters. Instead it appears toconverge to a possible local minimum. We demonstrate this fact by plotting the leastsquares objective function for the case of added constant variance noise correspondingto ν = 0.005 (1% RN) in the Figure 83. This figure clearly demonstrates the presenceof many local optima. The reason for the presence of local optima is the fact thatwe are changing ε∞,0 inside the Debye media. This changes the speed of propagationof the electromagnetic pulse inside the Debye media as the maximum wavespeed inthe dispersive dielectric is c0/

√ε∞,0 [Pet94]. A similar phenomenon was observed in

[BGW03] where the authors constructed a modified least squares objective functionto avoid the occurence of local optima. Another option to solve this problem would beto resort to using a global optimization method to solve the inverse problem. We notethat the Levenberg-Marquardt method did not seem to converge to a local minima.However, this may be a fortunate occurence in which case we can always find someinitial guesses that would cause such local optimization algorithms to converge to alocal minimum instead of the global minimum.

In Table 27 we calculate the confidence intervals for the two parameter estimationproblem. We note that the intervals are larger as compared with the results of theLevenberg-Marquardt method. We also plot the least squares objective functionfor different values of ε∞,0 and κ∞ without adding noise to the data. Again we seethe presence of local optima as seen in Figures 84. To see how the interrogatingfrequency relates to the presence of local minima we repeat the inverse problem bylowering the interrogating frequency to 3.0 GHz from 5.0 GHz. In Figure 85 Weobserve that in this case the distance between successive minima has increased. Thusby lowering the interrogating frequency we can push the local minima away from eachother and thius reduce the number of local minima that would occur in an interval ofinterest. We next attempt to estimate the three parameters τ0, ε∞,0 and its pressure

104

J S

5.3

5.5

5.7

5.9

1.2

1.4

1.6

1.8

20

100

200

300

400

ε∞,0

Least Squares Objective Function

κ∞

5.3 5.5 5.7 5.90

50

100

150

200

250

ε∞,0

JS

1.4 1.6 1.8 20

5

10

15

κ∞

JS

Least Squares Objective Function

Figure 79: (Top) The Least Squares Objective Function for different values of ε∞,0 andκ∞ with no noise added to the data. The other five parameters were fixed at valuesgiven in (116), and (117) except for τ0 which was fixed at its true value. The minimumis located at the point (5.5, 1.617), with a minimum value of 0.0147. (Bottom) CrossSections of the Least Squares Objective function versus κ∞ (right) and versus ε∞,0

(left).

105

5.3

5.5

5.7

5.9

1.2

1.4

1.6

1.8

20

100

200

300

400

JS

ε∞,0

Least Squares Objective Function

κ∞

5.3 5.5 5.7 5.90

50

100

150

200

250

ε∞,0

JS

1.4 1.6 1.8 20

4

8

12

16

20

κ∞

JS

Least Squares Objective Function Cross−sections

Figure 80: (Top) The Least Squares Objective Function for different values of ε∞,0

and κ∞ with no noise added to the data. The other five parameters were fixed atvalues given in (116), and (117), and τ0 fixed at the value given in (118) that have10-50% relative error as compared with their true values. The minimum is located atthe point (5.5, 1.584), with a minimum value of 0.03595. (Bottom) Cross Sections ofthe Least Squares Objective function versus κ∞ (right) and versus ε∞,0 (left).

106

0 10 20 30 40 5010−4

10−2

100

102

104L∞ Norm of Gradient

0 10 20 30 40 5010−10

10−5

100

105Function Differences

0 10 20 30 40 50102

103

104Least Squares Function Value

0 10 20 30 40 5010−6

10−4

10−2

100Max Oriented Length

0 10 20 30 40 5010−2

100

102

104L2 Norm of Gradient

Iterations

0 10 20 30 40 50102

103

104Average Least Squares value

Iterations

Figure 81: Details of the Nelder-Mead algorithms for the two parameter estimationof ε∞,0 and κ∞ for varying levels (ν) of constant variance noise added to the data astabulated in Table 26.

107

0 10 20 30 40 504.5

4.6

4.7

4.8

Iterations

ε ∞,0

^

ε∞,0* = 5.5

0 10 20 30 40 501

1.1

1.2

1.3

1.4

1.5

Iterations

κ^ ∞

κ∞* = 1.65

Figure 82: Variation in the estimates ε∞,0 (left) and κ∞

(right) over all iterations in the Nelder-Mead algorithmfor varying levels (ν) of constant variance noise addedto the data as tabulated in Table 26.

Table 26: Parameter estimation of ε∞,0 and κ∞ for varying levels of constant variancenoise corresponding to 0%-10% relative noise.

% RN Iter ε∞,0 κ∞ |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.1 39 4.5226 1.1231 8.5551 × 10−9 113.3545 0.0551 790.3 40 4.5224 1.1232 3.6791 × 10−9 114.9105 0.0440 810.5 42 4.5221 1.1233 1.6051 × 10−9 118.6662 0.0209 851.0 42 4.5215 1.1236 8.8755 × 10−9 137.6788 0.0610 833.0 42 4.5189 1.1246 5.7953 × 10−9 351.2076 0.0154 835.0 40 4.5164 1.1257 9.4709 × 10−9 784.7016 0.0439 7910.0 42 4.5102 1.1284 3.7162 × 10−9 2830.7864 0.0448 86

108

Table 27: Confidence Intervals for the parameter estima-tion of ε∞,0 and κ∞ with the Nelder-Mead algorithm forvarying levels of constant variance noise correspondingto 0%-10% relative noise.

RN = 0.1%ε∞,0 4.5226 ± 7.6273 × 10−3

κ∞ 1.1231 ± 4.7312 × 10−2

RN = 0.3%ε∞,0 4.5224 ± 7.6781 × 10−3

κ∞ 1.1232 ± 4.7624 × 10−2

RN = 0.5%ε∞,0 4.5221 ± 7.8008 × 10−3

κ∞ 1.1233 ± 4.8383 × 10−2

RN = 1.0%ε∞,0 4.5215 ± 8.3986 × 10−3

κ∞ 1.1236 ± 5.2073 × 10−2

RN = 3.0%ε∞,0 4.5189 ± 1.3385 × 10−2

κ∞ 1.1246 ± 8.2935 × 10−2

RN = 5.0%ε∞,0 4.5164 ± 1.9965 × 10−2

κ∞ 1.1257 ± 1.2361 × 10−1

RN = 10.0%ε∞,0 4.5102 ± 3.7722 × 10−2

κ∞ 1.1284 ± 2.3306 × 10−1

109

44.5

55.5

66.5

7

0.5

1

1.5

2

2.50

200

400

600

800

1000

1200

ε∞,0

κ∞

JS

Least Squares Objective Function (ν = 0.005)

4 4.5 5 5.5 6 6.5 70

100

200

300

400

500

ε∞,0

JS

0.8 1.2 1.6 2 2.420

40

60

80

100

κ∞

JS

Least Squares Objective Function Cross−sections

Figure 83: (Top) The Least Squares Objective Function for different values of ε∞,0

and κ∞ for constant variance noise with ν = 0.005 (1% RN)added to the data. Theother five parameters were fixed at values given in (116), and (117). The minimumis located at the point (5.5, 1.584), with a minimum value of 27.5152. However, wenote the presence of a local minimum at the point (4.51, 1.122) with a minimum leastsquares value of 138.0849. (Bottom) Cross sections of the Least Squares Objectivefunction versus κ∞ (right) and versus ε∞,0 (left).

110

2.5 3.5 4.5 5.5 6.5 7.5 8.5 01

23

0

1000

2000

3000

4000

5000

κ∞

Least Squares Objective Function

ε∞,0

JS

3 4 5 6 7 80

500

1000

1500

ε∞,0

JS

0 1 2 30

100

200

300

κ∞

JS

Least Squares Objective Function

Figure 84: (Top) The Least Squares Objective Function for different values of ε∞,0

and κ∞ with no noise added to the data. The other five parameters were fixed atvalues given in (116), and (117), and τ0 fixed at the value in (118). The minimumis located at the point (5.5, 1.58), with a minimum value of 0.036. However, we notethe presence of many local minimum. (Bottom) Cross Sections of the Least SquaresObjective function versus κ∞ (right) and versus ε∞,0 (left).

111

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 1.5

2

2.5

3

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

JS

k ∞

ε∞,0

Least Squares Objective Function

3 4 5 6 7 80

1000

2000

3000

4000

5000

ε∞,0

JS

1.5 2 2.5 30

100

200

300

400

500

600

κ∞

JS

Least Squares Objective Function: Cross−Sections

Figure 85: (Top) The Least Squares Objective Function for different values of ε∞,0

and κ∞ with no noise added to the data. The other five parameters were fixed atvalues given in (116), and (117). However the electromagnetic input source had acenter frequency of 3.0 GHz as opposed to 5.0 GHz. The minimum is located at thepoint (5.5, 1.683), with a minimum value of 6.91. We again note the presence of localminimum. (Bottom) Cross sections of the Least Squares Objective function versusκ∞ (right) and versus ε∞,0 (left).

112

Table 28: Parameter estimation of εs,0, τ0, and κs for 0%, 1%, 3% and 5% constantvariance noise (test 2).

% RN Iter ε∞,0 τ0(×10−8) κ∞

0.0 135 4.5306 1.3561 1.64700.1 120 4.5306 1.3475 1.65590.3 124 4.5301 1.3574 1.64790.5 122 4.5300 1.3875 1.62671.0 126 4.5283 1.4698 1.57873.0 118 4.5234 1.7950 1.42315.0 125 4.5185 2.4188 1.2588

Table 29: Parameter estimation of εs,0, τ0, and κs for 0%, 1%, 3% and 5% constantvariance noise (test 2).

% RN Iter |JSn+1 − JS

0 | JS ||∇JS||L2 FC0.0 135 7.3419 × 10−9 112.2835 0.0281 2440.1 120 5.7082 × 10−9 112.2681 0.0355 2160.3 124 6.8767 × 10−9 113.8858 0.0023 2260.5 122 2.4453 × 10−9 117.7016 0.0287 2171.0 126 8.1947 × 10−9 136.8571 0.0159 2253.0 118 9.0603 × 10−9 350.8494 0.0205 2065.0 125 4.3675 × 10−9 784.6225 0.0296 231

coefficient κ∞ again by adding constant variance noise to our data. The other fourparameters are fixed at the values given in (116), and (117). Table 28 displays theresults of the inverse problem. Again we note that the Nelder-Mead converges to alocal minima.Figures 86, 87, 88 and 89 plot the details of the Nelder-Mead algorithmand the variation of the parameters with the iteration count. In Table 30 we calculatethe confidence intervals for the three parameter estimation problem. Again we observethat the intervals are larger as compared with the results of the Levenberg-Marquardtmethod.

8 Conclusions and future directions

In this report we have presented an electromagnetic interrogation technique for iden-tifying dielectric parameters of a Debye medium by using acoustic pressure waves

113

Table 30: Confidence Intervals for the parameter esti-mation of ε∞,0, τ0 and κ∞ for constant variance noisewith the Nelder-Mead algorithm.

RN = 0.0%ε∞,0 4.5306 ± 8.2512 × 10−3

τ0 (1.3561 ± 5.9711 × 10−1) × 10−8

κ∞ 1.6470 ± 1.9556 × 10−1

RN = 0.1%ε∞,0 4.5306 ± 8.2283 × 10−3

τ0 (1.3475 ± 5.9323 × 10−1) × 10−8

κ∞ 1.6559 ± 1.9523 × 10−1

RN = 0.3%ε∞,0 4.5301 ± 8.2937 × 10−3

τ0 (1.3574 ± 6.0170 × 10−1) × 10−8

κ∞ 1.6479 ± 1.9682 × 10−1

RN = 0.5%ε∞,0 4.5300 ± 8.4345 × 10−3

τ0 (1.3875 ± 6.2477 × 10−1) × 10−8

κ∞ 1.6267 ± 2.0063 × 10−1

RN = 1.0%ε∞,0 4.5283 ± 9.0550 × 10−3

τ0 (1.4698 ± 7.1148 × 10−8) × 10−8

κ∞ 1.5787 ± 2.1736 × 10−1

RN = 3.0%ε∞,0 4.5234 ± 1.4519 × 10−2

τ0 (1.7950 ± 1.3804) × 10−8

κ∞ 1.4231 ± 3.5550 × 10−1

RN = 5.0%ε∞,0 4.5185 ± 2.1562 × 10−2

τ0 (2.4188 ± 2.7552) × 10−8

κ∞ 1.2588 ± 5.4570 × 10−1

114

0 20 40 60 80 100 120 14010−4

10−2

100

102

104

Iterations

L∞ Norm of Gradient

0 20 40 60 80 100 120 14010−10

10−5

100

105Function Differences

0 20 40 60 80 100 120 140

125.8925

251.1886

501.1872

Least Squares Function Value

0 20 40 60 80 100 120 14010−5

100Max Oriented Length

0 20 40 60 80 100 120 14010−4

10−2

100

102

104

Iterations

L2 Norm of Gradient

0 20 40 60 80 100 120 140102

103

104Average Least Squares value

Figure 86: Details of the Nelder-Mead algorithms for the three parameter estimationof ε∞,0, τ0 and κ∞ with varying levels (ν) of constant variance noise added to thedata as tabulated in Table 28, and 29.

115

0 20 40 60 80 100 120 1404.4

4.5

4.6

4.7

4.8

4.9

5

Iterations

ε ∞, 0

^

ε∞,0* = 5.5

Figure 87: Variation in the estimate ε∞,0 over all iterations in the Nelder-Mead algo-rithm for varying levels (ν) of constant variance noise added to the data as tabulatedin Table 28, and 29.

0 20 40 60 80 100 120 1401

1.5

2

2.5

3

3.5x 10−8

Iterations

τ

τ0

0^

* = 3.16× 10−8

Figure 88: Variation in the estimate τ0 over all iterations in the Nelder-Mead algo-rithm for varying levels (ν) of constant variance noise added to the data as tabulatedin Table 28, and 29.

116

0 20 40 60 80 100 120 1401

1.2

1.4

1.6

1.8

Iterations

κ^ ∞

κ∞* = 1.65

Figure 89: Variation in the estimate κ∞ over all iterations in the Nelder-Mead algo-rithm for varying levels (ν) of constant variance noise added to the data as tabulatedin Table 28, and 29.

as virtual reflectors for the incident electromagnetic pulse. We considered the twodimensional TE mode of Maxwell’s equations which incorporates the pressure de-pendence in the Debye medium via a model for orientational polarization. As a firstapproximation we assumed that the dielectric parameters εs,0, ε∞,0 and the relaxationτ0 are affine functions of pressure. We have used the finite difference time domainscheme to discretize our electromagnetic/acoustic model and to compute simulateddata to be used in a parameter identification problem.

We formulated an inverse problem for the identification of the material parametersas well as the coefficients of pressure in the Debye model, based on the method ofleast squares. We used two different methods to solve the inverse problem; namelyLevenberg-Marquardt and Nelder-Mead. The inverse problem indicates that the co-efficients that can be identified depend on the order of magnitude of ωτ0. Thus inthe first test problem we were able to identify εs,0 and the corresponding pressurecoefficient κs, and in the second test problem we were able to identify ε∞,0 and thecorresponding pressure coefficient κ∞. It is not possible to accurately identify allthe seven parameters that describe the Debye medium coupled with the model forpressure dependence in either of the examples that we have considered in this report.

We have also computed confidence intervals for all the estimates obtained usingstatistical error analysis based on the assumption of constant variance noise. Weshowed that the intervals become larger as the level of added noise is increased.

117

There are many questions that still need to be answered. We have used a simplelinear model for the pressure term in Maxwell’s equations. Also we have assumed thatthe effect of the electromagnetic field on the acoustic wave is negligible. A dynamiccoupled model of the electromagnetic/acoustic interaction is the topic of a futurepaper. We will also consider the dielectric parameters to be nonlinear functions ofpressure in a future paper to determine if the dynamics of the problem change withthe dependence of the dielectric parameters on the pressure.

9 Acknowledgements

This research was supported in part by the U.S. Air Force office of Scientific Researchunder grants AF0SR F49620-01-1-0026 and AFOSR FA9550-04-1-0220. The authorswould like to thank Dr. Richard Albanese of the AFRL, Brooks AFB, Brian Adams,Nathan Gibson, Dr. Grace Kepler, Dr. Yanyuan Ma, Dr. Jari Toivanen of CRSC atNCSU for their valuable comments and suggestions.

References

[ABK04] R. Albanese, H. T. Banks, and G. M. Kepler. Experimental results involv-ing acousto-optical interactions. Technical Report In preparation, CRSC,North Carolina State University, 2004.

[ABR02] R. A. Albanese, H. T. Banks, and J. K. Raye. Nondestructive evaluationof materials usig pulsed microwave interrogating signals and acoustic waveinduced reflections. Inverse Problems, 18:1935–1958, 2002.

[AMP94] R. A. Albanese, R. L. Medina, and J. W. Penn. Mathematics, medicineand microwaves. Inverse Problems, 10:995–1007, 1994.

[And67] J. C. Anderson. Dielectrics. Chapman and Hall LTD and Science Paper-backs, 1967.

[APM89] R. A. Albanese, J. W. Penn, and R. L. Medina. Short-rise-time microwavepulse propagation through dispersive biological media. J. Optical Societyof America A, 6:1441–1446, 1989.

[BB99] H. T. Banks and K. L. Bihari. Modeling and estimating uncertainty inparameter estimation. Technical Report CRSC-TR99-40, CRSC, NorthCarolina State University; Inverse Problems 17: 95-111, 2001, December1999.

[BB03] H. T. Banks and J. M. Bardsley. Parameter identification for a dispersivedielectric in 2d electromagnetics: Forward and inverse methodology with

118

statistical considerations. CRSC-TR03-48, December 2003; Intl. J. Comp.and Num. Anal. and Applic., to appear, 2003.

[BBL00] H. T. Banks, M. W. Buksas, and T. Lin. Electromagnetic Material Inter-rogation Using Conductive Interfaces and Acoustic Wavefronts. Frontiersin Applied Mathematics. SIAM, Philadelphia, PA, 2000.

[Ber94] J. P. Berenger. A perfectly matched layer for the absorption of electro-magnetic waves. J. Comput. Phys., 114(2):185–200, 1994.

[BF95] J. G. Blaschak and J. Franzen. Precursor propagation in dispersive me-dia from short-rise-time pulses at oblique incidence. J. Optical Society ofAmerica A, 12:1501–1512, 1995.

[BG04] H. T. Banks and N. L. Gibson. Well-posedness in maxwell systems with dis-tributions of polarization relaxation parameters. Technical Report CRSC-TR04-01, CRSC, North Carolina State University; Applied Math. Letters,to appear, 2004.

[BGW03] H. T. Banks, N. L. Gibson, and W. P. Winfree. Electromagnetic crack de-tection inverse problems using terahertz interrogating propagating signals.Technical Report CRSC-TR03-40, CRSC, North Carolina State University,October 2003.

[BK89] H. T. Banks and K. Kunisch. Estimation Techniques for Distributed Pa-rameter Systems. Birkhauser, Boston, 1989.

[Bri60] L. Brillouin. Wave Propagation and Group Velocity. Pure Appl. Phys.Academic Press, New York and London, 1960.

[DG95] M. Davidian and D. M. Giltinan. Nonlinear Models for Repeated Measure-ment Data, volume 62 of Monographs on Statistics and Applied probability.Chapman and Hall, New York, 1995.

[FMS03] E. C. Fear, P. M. Meaney, and M. A. Stuchly. Microwaves for breast cancerdetection. IEEE Potentials, pages 12–18, 2003.

[Ged96] S. D. Gedney. An anisotropic perfectly matched layer absorbing mediafor the truncation of FDTD lattices. IEEE Trans. Antennas Propagat.,44(12):1630–1639, 1996.

[Jac99] J. D. Jackson. Classical Electromagnetics. John Wiley and Sons, 1999.

[JN84] N. S. Jayant and P. Noll. Digital Coding of Waveforms. Prentice Hall,Englewoods Cliffs, 1984.

119

[Kel99] C. T. Kelley. Iterative Methods for Optimization. Frontiers in AppliedMathematics. SIAM, Philadelphia, PA, 1999.

[Kor97] A. Korpel. Acosto-Optics. Marcel-Dekker, New York, 2nd edition, 1997.

[MI68] P. Morse and K. Ingard. Theoretical Acoustics. McGraw-Hill, 1968.

[MP95] R. Mittra and U. Pekel. A new look at the perfectly matched layer (PML)concept for the reflectionless absorption of electromagnetic waves. IEEEMicrowave Guided Wave Lett., 5(3):84–86, 1995.

[Pet94] P. Petropoulos. Stability and phase error analysis of FD-TD in dispersivedielectrics. IEEE Trans. Antennas Propagat., 42(1):62–69, 1994.

[Rap95] C. M. Rappaport. Perfectly matched absorbing boundary conditions basedon anisotropic lossy mapping of space. IEEE Microwave Guided WaveLett., 5(3):90–92, 1995.

[SKLL95] Z. S. Sacks, D. M. Kinsland, R. Lee, and J. F. Lee. A perfectly matchedanisotropic absorber for use as an absorbing boundary condition. IEEETrans. Antennas Propagat., 43:1460–1463, 1995.

[Taf95] A. Taflove. Computational Electrodynamics: The Finite-Difference Time-Domain method. Artech House, Norwood, MA, 1995.

[Taf98] A. Taflove. Advances in Computational Electrodynamics: The Finite-Difference Time-Domain method. Artech House, Norwood, MA, 1998.

120